{"id":19874,"date":"2022-02-23T01:32:13","date_gmt":"2022-02-23T01:32:13","guid":{"rendered":"https:\/\/onlinemathtutor.co\/?page_id=19874"},"modified":"2022-05-19T20:51:09","modified_gmt":"2022-05-19T20:51:09","slug":"integration-practice-problems-partial-fractions","status":"publish","type":"page","link":"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-partial-fractions\/","title":{"rendered":"Integration Practice Problems &#8211; [Partial Fractions]"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"19874\" class=\"elementor elementor-19874\" data-elementor-post-type=\"page\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-122add1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"122add1\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f69b248\" data-id=\"f69b248\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fba6159 elementor-widget elementor-widget-heading\" data-id=\"fba6159\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Practice Problems: Partial Fractions<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9ac37bb elementor-nav-menu__align-center elementor-nav-menu--stretch elementor-nav-menu__text-align-center animated-fast elementor-nav-menu--dropdown-tablet elementor-nav-menu--toggle elementor-nav-menu--burger elementor-widget elementor-widget-nav-menu\" data-id=\"9ac37bb\" data-element_type=\"widget\" data-settings=\"{&quot;submenu_icon&quot;:{&quot;value&quot;:&quot;&lt;i class=\\&quot;fas fa-pencil-alt\\&quot;&gt;&lt;\\\/i&gt;&quot;,&quot;library&quot;:&quot;fa-solid&quot;},&quot;full_width&quot;:&quot;stretch&quot;,&quot;_animation&quot;:&quot;none&quot;,&quot;layout&quot;:&quot;horizontal&quot;,&quot;toggle&quot;:&quot;burger&quot;}\" data-widget_type=\"nav-menu.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<nav aria-label=\"Menu\" class=\"elementor-nav-menu--main elementor-nav-menu__container elementor-nav-menu--layout-horizontal e--pointer-double-line e--animation-fast\">\n\t\t\t\t<ul id=\"menu-1-9ac37bb\" class=\"elementor-nav-menu\"><li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-19677\"><a href=\"https:\/\/onlinemathtutor.co\/index.php\/problem-set-integration\/\" class=\"elementor-item\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2021\/06\/Home-Icon-Blue.png\"  width=\"20\"><\/a><\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19577\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Integration Techniques<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19777\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">U-Substitution<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19576\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-usub\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19776\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-usub\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-19694\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-ibp\/\" class=\"elementor-sub-item\">Integration By Parts<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19898\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-ibp\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19956\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-ibp\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19885\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Trigonometric Integrals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19925\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-integrals\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19954\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-integrals\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19886\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Trigonometric Substitution<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19926\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-substitution\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19953\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-substitution\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19884\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Partial Fractions<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19924\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-partial-fractions\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19955\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-partialfractions\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19893\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Improper Integrals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19927\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-improper-integrals\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19966\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-improper-integrals\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19578\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Integration Applications<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19888\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Area<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19928\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-area\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20136\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-area\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19889\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Volumes [Nonrotational]<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19929\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-volume\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20582\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-substitution\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19890\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Volume [Rotational]<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19931\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-rotational-volumes\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20160\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-rotational-volume\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19891\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Surface Area<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19930\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-surface-area\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20165\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-surface-area\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19892\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Arc Length<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20177\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-arc-length\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20176\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-arc-length\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19579\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Numerical Integration<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19894\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Reimann Sums<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20110\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-riemann-sums\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20128\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-riemann-sum\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19895\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Trapezoidal Rule<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20109\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trapezoidal\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20138\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trapezoidal\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19896\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Simpson&#8217;s Rule<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20108\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-simpsons\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20137\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-simpsons\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t<div class=\"elementor-menu-toggle\" role=\"button\" tabindex=\"0\" aria-label=\"Menu Toggle\" aria-expanded=\"false\">\n\t\t\t<i aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--open eicon-menu-bar\"><\/i><i aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--close eicon-close\"><\/i>\t\t<\/div>\n\t\t\t\t\t<nav class=\"elementor-nav-menu--dropdown elementor-nav-menu__container\" aria-hidden=\"true\">\n\t\t\t\t<ul id=\"menu-2-9ac37bb\" class=\"elementor-nav-menu\"><li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-19677\"><a href=\"https:\/\/onlinemathtutor.co\/index.php\/problem-set-integration\/\" class=\"elementor-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2021\/06\/Home-Icon-Blue.png\"  width=\"20\"><\/a><\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19577\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Integration Techniques<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19777\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">U-Substitution<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19576\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-usub\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19776\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-usub\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-19694\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-ibp\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Integration By Parts<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19898\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-ibp\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19956\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-ibp\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19885\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Trigonometric Integrals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19925\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-integrals\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19954\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-integrals\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19886\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Trigonometric Substitution<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19926\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-substitution\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19953\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-substitution\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19884\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Partial Fractions<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19924\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-partial-fractions\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19955\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-partialfractions\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19893\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Improper Integrals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19927\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-improper-integrals\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19966\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-improper-integrals\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19578\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Integration Applications<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19888\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Area<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19928\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-area\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20136\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-area\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19889\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Volumes [Nonrotational]<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19929\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-volume\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20582\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-substitution\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19890\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Volume [Rotational]<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19931\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-rotational-volumes\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20160\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-rotational-volume\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19891\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Surface Area<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19930\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-surface-area\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20165\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-surface-area\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19892\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Arc Length<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20177\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-arc-length\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20176\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-arc-length\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19579\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Numerical Integration<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19894\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Reimann Sums<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20110\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-riemann-sums\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20128\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-riemann-sum\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19895\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Trapezoidal Rule<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20109\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trapezoidal\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20138\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trapezoidal\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19896\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Simpson&#8217;s Rule<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20108\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-simpsons\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20137\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-simpsons\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a75a15d elementor-section-stretched elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a75a15d\" data-element_type=\"section\" data-settings=\"{&quot;stretch_section&quot;:&quot;section-stretched&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-555d8f5\" data-id=\"555d8f5\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-356b90a elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"356b90a\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6f95eaf elementor-widget elementor-widget-text-editor\" data-id=\"6f95eaf\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"text-align: justify;\"><span style=\"font-size: 20px;\">Although partial fraction decomposition (or simply partial fractions) is an algebraic, not an integration technique, it is useful for integrating rational functions. When decomposing the ration function, remember to echo any repeated factors. A rational functions has the form \\(\\textstyle f(x)=\\frac{P(x)}{Q(x)}\\) where \\(P(x)\\) and \\(Q(x)\\) are both polynomials<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-40880ff elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"40880ff\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a99c65d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a99c65d\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-e0f1a78\" data-id=\"e0f1a78\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-fb6564e\" data-id=\"fb6564e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-48678aa elementor-view-framed elementor-position-left elementor-shape-circle elementor-mobile-position-top elementor-vertical-align-top elementor-widget elementor-widget-icon-box\" data-id=\"48678aa\" data-element_type=\"widget\" data-widget_type=\"icon-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-icon-box-wrapper\">\n\n\t\t\t\t\t\t<div class=\"elementor-icon-box-icon\">\n\t\t\t\t<a href=\"https:\/\/onlinemathtutor.co\/index.php\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-partialfractions\/\" class=\"elementor-icon elementor-animation-grow\" tabindex=\"-1\" aria-label=\"Virtual Lessons\">\n\t\t\t\t<i aria-hidden=\"true\" class=\"fas fa-chalkboard-teacher\"><\/i>\t\t\t\t<\/a>\n\t\t\t<\/div>\n\t\t\t\n\t\t\t\t\t\t<div class=\"elementor-icon-box-content\">\n\n\t\t\t\t\t\t\t\t\t<h2 class=\"elementor-icon-box-title\">\n\t\t\t\t\t\t<a href=\"https:\/\/onlinemathtutor.co\/index.php\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-partialfractions\/\" >\n\t\t\t\t\t\t\tVirtual Lessons\t\t\t\t\t\t<\/a>\n\t\t\t\t\t<\/h2>\n\t\t\t\t\n\t\t\t\t\t\t\t\t\t<p class=\"elementor-icon-box-description\">\n\t\t\t\t\t\tNeed some additional help understanding how to apply this integration technique? Click Here to visit the virtual lesson section. \t\t\t\t\t<\/p>\n\t\t\t\t\n\t\t\t<\/div>\n\t\t\t\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-9bfaa1b\" data-id=\"9bfaa1b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b1c245c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b1c245c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-db446e9\" data-id=\"db446e9\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-86d2d68 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"86d2d68\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b9b6a13 elementor-widget elementor-widget-spacer\" data-id=\"b9b6a13\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7c20c1d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7c20c1d\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6c836f6\" data-id=\"6c836f6\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f33923c elementor-widget elementor-widget-heading\" data-id=\"f33923c\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-large\">Practice Problems<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8562456 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8562456\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a9bedd6\" data-id=\"a9bedd6\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0026ba1 elementor-widget elementor-widget-text-editor\" data-id=\"0026ba1\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"text-align: justify;\"><span style=\"font-size: 18px;\">Evaluate the following questions by using technique of partial Partial Fraction Decomposition.\u00a0<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-26e78c3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"26e78c3\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-b97601e\" data-id=\"b97601e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-164107e elementor-widget elementor-widget-text-editor\" data-id=\"164107e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ol start=\"1\"><li style=\"margin-bottom: 30px;\">\\(\\displaystyle f(x) = \\frac{2}{x^2+6x+8} \\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle f(x) = \\frac{1}{x(x+1)(x+2)^2} \\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{dx}{x^2-4} \\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{x+1}{2x^2+7x+3} dx \\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{3x+1}{x^2+5x+6} dx \\)<\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-a6153c1\" data-id=\"a6153c1\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0036d76 elementor-widget elementor-widget-text-editor\" data-id=\"0036d76\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ol start=\"6\">\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{1-2x}{x^3+4x^2+4x} dx \\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{x-1}{x(x^2+1)} dx \\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{x}{x^2+2x+10} dx \\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{\\sqrt{x+1}}{x} dx \\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{\\sqrt{x+1}}{\\sqrt{x-1}} dx \\)<\/li>\n<\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-0282234\" data-id=\"0282234\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3a06027 elementor-widget elementor-widget-text-editor\" data-id=\"3a06027\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ol start=\"11\"><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{d\\theta}{1+\\texttt{sin}(\\theta)}\u00a0 \\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{d\\theta}{1+\\texttt{sin}(\\theta)+\\texttt{cos}(\\theta)}\u00a0 \\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{\\texttt{sin}(\\theta)}{\\texttt{cos}(\\theta) + \\texttt{sin}(\\theta)} d\\theta \\)<\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-21017cc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"21017cc\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1c0c34c\" data-id=\"1c0c34c\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8ceb276 elementor-align-left elementor-icon-list--layout-traditional elementor-list-item-link-full_width elementor-widget elementor-widget-icon-list\" data-id=\"8ceb276\" data-element_type=\"widget\" data-widget_type=\"icon-list.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul class=\"elementor-icon-list-items\">\n\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<a href=\"https:\/\/onlinemathtutor.co\/index.php\/problem-set-integration\/\">\n\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<i aria-hidden=\"true\" class=\"fas fa-arrow-circle-right\"><\/i>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Back To Main Page: Integration Problem Sets<\/span>\n\t\t\t\t\t\t\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t<\/ul>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b7f6824 elementor-widget elementor-widget-spacer\" data-id=\"b7f6824\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9d0490a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9d0490a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-09b8a9e\" data-id=\"09b8a9e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9422751 elementor-widget elementor-widget-heading\" data-id=\"9422751\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-large\">Suggestive Solution Guide<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6b529fe9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6b529fe9\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3e5bd0e\" data-id=\"3e5bd0e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5081f69d elementor-widget elementor-widget-toggle\" data-id=\"5081f69d\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1351\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1351\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 1: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1351\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1351\"><h3>This exercise aims to show how the partial fractions decomposition works:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\frac{2}{x^2+6x+8}<br \/>&amp;= \\frac{2}{(x+4)(x+2)} = \\frac{A}{x+4}\\:+ \\frac{B}{x+2}<br \/>&amp; \\color{gray}{\\text{[Factor the denominator and separte the fraction]}} \\\\[5pt]<br \/>&amp; \\left( \\frac{2}{(x+4)(x+2)} = \\frac{A}{x+4}\\:+ \\frac{B}{x+2} \\right)\u00a0 (x+4)(x+2)<br \/>&amp; \\color{gray}{\\text{[Multiply the equation by the factored denominator]}} \\\\[5pt]<br \/>&amp; 2 = A(x+2)\\:+ B(x+4)<br \/>&amp; \\color{gray}{\\text{[Main equation to solve]}} \\\\[5pt]<br \/>&amp; 2 = A(-4+2)\\:+ B(-4+4)<br \/>&amp; \\color{gray}{\\text{[If x = -4]}} \\\\[5pt]<br \/>&amp; 2 = -2A \\\\[5pt]<br \/>&amp; \\color{blue}{A = -1} \\\\\\\\<br \/>&amp; 2 = A(x+2)\\:+ B(x+4)<br \/>&amp; \\color{gray}{\\text{[Main equation to solve]}} \\\\[5pt]<br \/>&amp; 2 = A(-2+2)\\:+ B(-2+4)<br \/>&amp; \\color{gray}{\\text{[If x = -2]}} \\\\[5pt]<br \/>&amp; 2 = 2B \\\\[5pt]<br \/>&amp; \\color{blue}{B = 1}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><h3>Replace the values of A and B in the partial fractions decomposition:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\frac{2}{x^2+6x+8} = \\color{green}{-\\frac{1}{x+4}\\:+ \\frac{1}{x+2}}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 1: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>This exercise aims to show how the partial fractions decomposition works:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\frac{2}{x^2+6x+8}<br \\\/>&amp;= \\\\frac{2}{(x+4)(x+2)} = \\\\frac{A}{x+4}\\\\:+ \\\\frac{B}{x+2}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Factor the denominator and separte the fraction]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\left( \\\\frac{2}{(x+4)(x+2)} = \\\\frac{A}{x+4}\\\\:+ \\\\frac{B}{x+2} \\\\right)\\u00a0 (x+4)(x+2)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Multiply the equation by the factored denominator]}} \\\\\\\\[5pt]<br \\\/>&amp; 2 = A(x+2)\\\\:+ B(x+4)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve]}} \\\\\\\\[5pt]<br \\\/>&amp; 2 = A(-4+2)\\\\:+ B(-4+4)<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x = -4]}} \\\\\\\\[5pt]<br \\\/>&amp; 2 = -2A \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{A = -1} \\\\\\\\\\\\\\\\<br \\\/>&amp; 2 = A(x+2)\\\\:+ B(x+4)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve]}} \\\\\\\\[5pt]<br \\\/>&amp; 2 = A(-2+2)\\\\:+ B(-2+4)<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x = -2]}} \\\\\\\\[5pt]<br \\\/>&amp; 2 = 2B \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{B = 1}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><h3>Replace the values of A and B in the partial fractions decomposition:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\frac{2}{x^2+6x+8} = \\\\color{green}{-\\\\frac{1}{x+4}\\\\:+ \\\\frac{1}{x+2}}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-288292b7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"288292b7\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-b5d9c06\" data-id=\"b5d9c06\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3dbf04df elementor-widget elementor-widget-toggle\" data-id=\"3dbf04df\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1031\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1031\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 2: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1031\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1031\"><h3>This exercise aims to show how the partial fractions decomposition works:<\/h3>\\(<br \/>\\begin{align*}<br \/>&amp; \\frac{1}{x(x+1)(x+2)^2} = \\frac{A}{x}\\:+ \\frac{B}{x+1}\\:+ \\frac{C}{x+2}\\:+ \\frac{D}{(x+2)^2}<br \/>&amp; \\color{gray}{\\text{[Denominator already factored]}} \\\\[5pt]<br \/>&amp; \\left( \\frac{1}{x(x+1)(x+2)^2} = \\frac{A}{x}\\:+ \\frac{B}{x+1}\\:+ \\frac{C}{x+2}\\:+ \\frac{D}{(x+2)^2} \\right) x(x+1)(x+2)^2<br \/>&amp; \\color{gray}{\\text{[Multiply the equation by the factors]}} \\\\[5pt]<br \/>&amp; 1 = A(x+1)(x+2)^2\\:+ Bx(x+2)^2\\:+ Cx(x+1)(x+2)\\:+ Dx(x+1)<br \/>&amp; \\color{gray}{\\text{[Main equation to solve]}} \\\\[5pt]<br \/>&amp; 1 = 4A\\:+ \\color{red}{0B}\\:+ \\color{red}{0C}\\:+ \\color{red}{0D}<br \/>&amp; \\color{gray}{\\text{[If x=0]}} \\\\[5pt]<br \/>&amp; \\color{blue}{A = \\frac{1}{4}} \\\\\\\\<br \/>&amp; 1 = A(x+1)(x+2)^2\\:+ Bx(x+2)^2\\:+ Cx(x+1)(x+2)\\:+ Dx(x+1)<br \/>&amp; \\color{gray}{\\text{[Main equation to solve]}} \\\\[5pt]<br \/>&amp; 1 = \\color{red}{0A}\\:- B\\:+ \\color{red}{0C}\\:+ \\color{red}{0D}<br \/>&amp; \\color{gray}{\\text{[If x=-1]}} \\\\[5pt]<br \/>&amp; \\color{blue}{B = -1} \\\\\\\\<br \/>&amp; 1 = A(x+1)(x+2)^2\\:+ Bx(x+2)^2\\:+ Cx(x+1)(x+2)\\:+ Dx(x+1)<br \/>&amp; \\color{gray}{\\text{[Main equation to solve]}} \\\\[5pt]<br \/>&amp; 1 = \\color{red}{0A}\\:+ \\color{red}{0B}\\:+ \\color{red}{0D}\\:+ 2D<br \/>&amp; \\color{gray}{\\text{[If x=-2]}} \\\\[5pt]<br \/>&amp; \\color{blue}{D = \\frac{1}{2}} \\\\\\\\<br \/>&amp; 1 = A(x+1)(x+2)^2\\:+ Bx(x+2)^2\\:+ Cx(x+1)(x+2)\\:+ Dx(x+1)<br \/>&amp; \\color{gray}{\\text{[Main equation to solve]}} \\\\[5pt]<br \/>&amp; 1 = 18A+9B+6C+2D<br \/>&amp; \\color{gray}{\\text{[If x=1]}} \\\\[5pt]<br \/>&amp; 1 = 18\\left(\\frac{1}{4}\\right)\\:+ 9(-1)\\:+ 6C\\:+ 2\\left(\\frac{1}{2}\\right) \\\\[5pt]<br \/>&amp; 1 = \\frac{9}{2}-9+6C+1 \\\\[5pt]<br \/>&amp; \\color{blue}{C = \\frac{3}{4}}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><h3>Update the partial fractions with the new found values:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\frac{1}{x(x+1)(x+2)^2} = \\color{green}{\\frac{1}{4x}\\:- \\frac{1}{x+1}\\:+ \\frac{3}{4(x+2)}\\:+ \\frac{1}{2(x+2)^2}}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 2: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>This exercise aims to show how the partial fractions decomposition works:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; \\\\frac{1}{x(x+1)(x+2)^2} = \\\\frac{A}{x}\\\\:+ \\\\frac{B}{x+1}\\\\:+ \\\\frac{C}{x+2}\\\\:+ \\\\frac{D}{(x+2)^2}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Denominator already factored]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\left( \\\\frac{1}{x(x+1)(x+2)^2} = \\\\frac{A}{x}\\\\:+ \\\\frac{B}{x+1}\\\\:+ \\\\frac{C}{x+2}\\\\:+ \\\\frac{D}{(x+2)^2} \\\\right) x(x+1)(x+2)^2<br \\\/>&amp; \\\\color{gray}{\\\\text{[Multiply the equation by the factors]}} \\\\\\\\[5pt]<br \\\/>&amp; 1 = A(x+1)(x+2)^2\\\\:+ Bx(x+2)^2\\\\:+ Cx(x+1)(x+2)\\\\:+ Dx(x+1)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve]}} \\\\\\\\[5pt]<br \\\/>&amp; 1 = 4A\\\\:+ \\\\color{red}{0B}\\\\:+ \\\\color{red}{0C}\\\\:+ \\\\color{red}{0D}<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x=0]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{A = \\\\frac{1}{4}} \\\\\\\\\\\\\\\\<br \\\/>&amp; 1 = A(x+1)(x+2)^2\\\\:+ Bx(x+2)^2\\\\:+ Cx(x+1)(x+2)\\\\:+ Dx(x+1)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve]}} \\\\\\\\[5pt]<br \\\/>&amp; 1 = \\\\color{red}{0A}\\\\:- B\\\\:+ \\\\color{red}{0C}\\\\:+ \\\\color{red}{0D}<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x=-1]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{B = -1} \\\\\\\\\\\\\\\\<br \\\/>&amp; 1 = A(x+1)(x+2)^2\\\\:+ Bx(x+2)^2\\\\:+ Cx(x+1)(x+2)\\\\:+ Dx(x+1)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve]}} \\\\\\\\[5pt]<br \\\/>&amp; 1 = \\\\color{red}{0A}\\\\:+ \\\\color{red}{0B}\\\\:+ \\\\color{red}{0D}\\\\:+ 2D<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x=-2]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{D = \\\\frac{1}{2}} \\\\\\\\\\\\\\\\<br \\\/>&amp; 1 = A(x+1)(x+2)^2\\\\:+ Bx(x+2)^2\\\\:+ Cx(x+1)(x+2)\\\\:+ Dx(x+1)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve]}} \\\\\\\\[5pt]<br \\\/>&amp; 1 = 18A+9B+6C+2D<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x=1]}} \\\\\\\\[5pt]<br \\\/>&amp; 1 = 18\\\\left(\\\\frac{1}{4}\\\\right)\\\\:+ 9(-1)\\\\:+ 6C\\\\:+ 2\\\\left(\\\\frac{1}{2}\\\\right) \\\\\\\\[5pt]<br \\\/>&amp; 1 = \\\\frac{9}{2}-9+6C+1 \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{C = \\\\frac{3}{4}}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><h3>Update the partial fractions with the new found values:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\frac{1}{x(x+1)(x+2)^2} = \\\\color{green}{\\\\frac{1}{4x}\\\\:- \\\\frac{1}{x+1}\\\\:+ \\\\frac{3}{4(x+2)}\\\\:+ \\\\frac{1}{2(x+2)^2}}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5a4268c9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5a4268c9\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1ab9073a\" data-id=\"1ab9073a\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-51024c94 elementor-widget elementor-widget-toggle\" data-id=\"51024c94\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1351\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1351\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 3: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1351\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1351\"><h3>Perform a partial fractions decomposition for the integrand:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{1}{x^2-4} dx<br \/>&amp; = \\int \\frac{1}{(x+2)(x-2)} dx = \\int \\frac{A}{x+2} dx\\:+ \\int \\frac{B}{x-2} dx<br \/>&amp; \\color{gray}{\\text{[Factor and separate the fraction]}}\\\\\\\\<br \/>&amp; 1 = A(x-2)+ B(x+2)<br \/>&amp; \\color{gray}{\\text{[Main equation to solve]}} \\\\[5pt]<br \/>&amp; 1 = -4A+ \\color{red}{0B}<br \/>&amp; \\color{gray}{\\text{[If x=- 2]}} \\\\[5pt]<br \/>&amp; \\color{blue}{A = -\\frac{1}{4}}\\\\\\\\<br \/>&amp; 1 = A(x-2)+ B(x+2)<br \/>&amp; \\color{gray}{\\text{[Main equation to solve]}} \\\\[5pt]<br \/>&amp; 1 = \\color{red}{0A}\\:+ 4B<br \/>&amp; \\color{gray}{\\text{[If x=2]}} \\\\[5pt]<br \/>&amp; \\color{blue}{B = \\frac{1}{4}}\\\\\\\\<br \/>\\end{align*}<br \/>\\)<h3>Update the numerator values for the partial fractions:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{1}{x^2-4}dx<br \/>&amp; = \\int \\frac{-1}{4(x+2)}dx\\:+ \\int \\frac{1}{4(x-2)}dx<br \/>&amp; \\color{gray}{\\text{[Update the values]}} \\\\[5pt]<br \/>&amp; = \\frac{1}{4}\\left[ \\int \\frac{1}{(x-2)}dx\\:- \\int \\frac{1}{(x+2)}dx \\right]<br \/>&amp; \\color{gray}{\\text{[Rearrange the expression]}} \\\\[5pt]<br \/>&amp; = \\frac{1}{4}\\left[ \\texttt{ln}|x-2| &#8211; \\texttt{ln}|x+2| \\right]\\:+ C<br \/>&amp; \\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp; \\color{green}{= \\frac{1}{4} \\texttt{ln} \\left| \\frac{x-2}{x+2} \\right|\\:+ C}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 3: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Perform a partial fractions decomposition for the integrand:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{1}{x^2-4} dx<br \\\/>&amp; = \\\\int \\\\frac{1}{(x+2)(x-2)} dx = \\\\int \\\\frac{A}{x+2} dx\\\\:+ \\\\int \\\\frac{B}{x-2} dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Factor and separate the fraction]}}\\\\\\\\\\\\\\\\<br \\\/>&amp; 1 = A(x-2)+ B(x+2)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve]}} \\\\\\\\[5pt]<br \\\/>&amp; 1 = -4A+ \\\\color{red}{0B}<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x=- 2]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{A = -\\\\frac{1}{4}}\\\\\\\\\\\\\\\\<br \\\/>&amp; 1 = A(x-2)+ B(x+2)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve]}} \\\\\\\\[5pt]<br \\\/>&amp; 1 = \\\\color{red}{0A}\\\\:+ 4B<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x=2]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{B = \\\\frac{1}{4}}\\\\\\\\\\\\\\\\<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Update the numerator values for the partial fractions:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{1}{x^2-4}dx<br \\\/>&amp; = \\\\int \\\\frac{-1}{4(x+2)}dx\\\\:+ \\\\int \\\\frac{1}{4(x-2)}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Update the values]}} \\\\\\\\[5pt]<br \\\/>&amp; = \\\\frac{1}{4}\\\\left[ \\\\int \\\\frac{1}{(x-2)}dx\\\\:- \\\\int \\\\frac{1}{(x+2)}dx \\\\right]<br \\\/>&amp; \\\\color{gray}{\\\\text{[Rearrange the expression]}} \\\\\\\\[5pt]<br \\\/>&amp; = \\\\frac{1}{4}\\\\left[ \\\\texttt{ln}|x-2| &#8211; \\\\texttt{ln}|x+2| \\\\right]\\\\:+ C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{= \\\\frac{1}{4} \\\\texttt{ln} \\\\left| \\\\frac{x-2}{x+2} \\\\right|\\\\:+ C}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-79f9f652 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"79f9f652\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-530f161c\" data-id=\"530f161c\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-35a604ff elementor-widget elementor-widget-toggle\" data-id=\"35a604ff\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-9001\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-9001\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 4: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-9001\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-9001\"><h3>Perform a partial fractions decomposition for the integrand:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{x+1}{2x^2+7x+3}dx<br \/>&amp; = \\int \\frac{x+1}{(2x+1)(x+3)}dx = \\int \\frac{A}{2x+1}dx\\:+ \\int \\frac{B}{x+3}dx \\\\\\\\<br \/>&amp; x+1 = A(x+3) + B(2x+1)<br \/>&amp; \\color{gray}{\\text{[Main equation to solve]}} \\\\[5pt]<br \/>&amp; \\frac{1}{2} = \\frac{5}{2}A + \\color{red}{0B}<br \/>&amp; \\color{gray}{\\text{[If x=0.5]}} \\\\[5pt]<br \/>&amp; \\color{blue}{A = \\frac{1}{5}}\\\\\\\\<br \/>&amp; x+1 = A(x+3) + B(2x+1)<br \/>&amp; \\color{gray}{\\text{[Main equation to solve]}} \\\\[5pt]<br \/>&amp; -2 = \\color{red}{0A} &#8211; 5B<br \/>&amp; \\color{gray}{\\text{[If x=-3]}} \\\\[5pt]<br \/>&amp; \\color{blue}{B = \\frac{2}{5}}\\\\\\\\<br \/>\\end{align*}<br \/>\\)<h3>Update the new values in the partial fractions:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{x+1}{2x^2+7x+3}dx<br \/>&amp; \\int \\frac {1}{5(2x+1)}dx + \\int \\frac{2}{5(x+3)}dx<br \/>&amp; \\color{gray}{\\text{[Update A and B]}} \\\\[5pt]<br \/>&amp; = \\frac{1}{5} \\left[ \\int \\frac{1}{2x+1}dx + 2\\int \\frac{1}{x+3}dx \\right]<br \/>&amp; \\color{gray}{\\text{[Rearrange the expression]}} \\\\[5pt]<br \/>&amp; = \\color{green}{\\frac{1}{5} \\left[ \\frac{1}{2}\\texttt{Ln}|2x+1| + 2\\texttt{Ln}|x+3| \\right] + C}<br \/>&amp; \\color{gray}{\\text{[Integrate using quick U-Subs]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 4: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Perform a partial fractions decomposition for the integrand:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{x+1}{2x^2+7x+3}dx<br \\\/>&amp; = \\\\int \\\\frac{x+1}{(2x+1)(x+3)}dx = \\\\int \\\\frac{A}{2x+1}dx\\\\:+ \\\\int \\\\frac{B}{x+3}dx \\\\\\\\\\\\\\\\<br \\\/>&amp; x+1 = A(x+3) + B(2x+1)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\frac{1}{2} = \\\\frac{5}{2}A + \\\\color{red}{0B}<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x=0.5]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{A = \\\\frac{1}{5}}\\\\\\\\\\\\\\\\<br \\\/>&amp; x+1 = A(x+3) + B(2x+1)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve]}} \\\\\\\\[5pt]<br \\\/>&amp; -2 = \\\\color{red}{0A} &#8211; 5B<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x=-3]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{B = \\\\frac{2}{5}}\\\\\\\\\\\\\\\\<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Update the new values in the partial fractions:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{x+1}{2x^2+7x+3}dx<br \\\/>&amp; \\\\int \\\\frac {1}{5(2x+1)}dx + \\\\int \\\\frac{2}{5(x+3)}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Update A and B]}} \\\\\\\\[5pt]<br \\\/>&amp; = \\\\frac{1}{5} \\\\left[ \\\\int \\\\frac{1}{2x+1}dx + 2\\\\int \\\\frac{1}{x+3}dx \\\\right]<br \\\/>&amp; \\\\color{gray}{\\\\text{[Rearrange the expression]}} \\\\\\\\[5pt]<br \\\/>&amp; = \\\\color{green}{\\\\frac{1}{5} \\\\left[ \\\\frac{1}{2}\\\\texttt{Ln}|2x+1| + 2\\\\texttt{Ln}|x+3| \\\\right] + C}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate using quick U-Subs]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3adffac0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3adffac0\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4408eb74\" data-id=\"4408eb74\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6b049f79 elementor-widget elementor-widget-toggle\" data-id=\"6b049f79\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1791\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1791\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 5: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1791\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1791\"><h3>Perform a partial fractions decomposition for the integrand:<\/h3>\\(\\normalsize<br \/>\\begin{align*}<br \/>\\int \\frac{3x+1}{x^2+5x+6}dx<br \/>&amp;= \\int \\frac{3x+1}{(x+3)(x+2)}dx = \\int \\frac{A}{x+3}dx + \\int \\frac{B}{x+2}dx\\\\\\\\<br \/>&amp; 3x+1 = A(x+2)+ B(x+3)<br \/>&amp; \\color{gray}{\\text{[Main equation to solve for]}} \\\\[5pt]<br \/>&amp; -8= -A + \\color{red}{0B}<br \/>&amp; \\color{gray}{\\text{[If x=-3]}} \\\\[5pt]<br \/>&amp; \\color{blue}{A = 8} \\\\\\\\<br \/>&amp; 3x+1 = A(x+2)+ B(x+3)<br \/>&amp; \\color{gray}{\\text{[Main equation to solve for]}} \\\\[5pt]<br \/>&amp; -5= \\color{red}{0A} + B<br \/>&amp; \\color{gray}{\\text{[If x=-2]}} \\\\[5pt]<br \/>&amp; \\color{blue}{B = -5}<br \/>\\end{align*}<br \/>\\)<h3>Update the values for the partial fractions:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{3x+1}{x^2+5x+6}dx<br \/>&amp;= \\int \\frac{8}{x+3}dx\\:- \\int \\frac{5}{x+2}dx<br \/>&amp; \\color{gray}{\\text{[Update the values for A and B]}} \\\\[5pt]<br \/>&amp; \\color{green}{= 8\\texttt{ln}|x+3|\\:- 5\\texttt{ln}|x+2|\\:+ C}<br \/>&amp; \\color{gray}{\\text{[Integrate using a quick U-Sub]}}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 5: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Perform a partial fractions decomposition for the integrand:<\\\/h3>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{3x+1}{x^2+5x+6}dx<br \\\/>&amp;= \\\\int \\\\frac{3x+1}{(x+3)(x+2)}dx = \\\\int \\\\frac{A}{x+3}dx + \\\\int \\\\frac{B}{x+2}dx\\\\\\\\\\\\\\\\<br \\\/>&amp; 3x+1 = A(x+2)+ B(x+3)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve for]}} \\\\\\\\[5pt]<br \\\/>&amp; -8= -A + \\\\color{red}{0B}<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x=-3]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{A = 8} \\\\\\\\\\\\\\\\<br \\\/>&amp; 3x+1 = A(x+2)+ B(x+3)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve for]}} \\\\\\\\[5pt]<br \\\/>&amp; -5= \\\\color{red}{0A} + B<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x=-2]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{B = -5}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Update the values for the partial fractions:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{3x+1}{x^2+5x+6}dx<br \\\/>&amp;= \\\\int \\\\frac{8}{x+3}dx\\\\:- \\\\int \\\\frac{5}{x+2}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Update the values for A and B]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{= 8\\\\texttt{ln}|x+3|\\\\:- 5\\\\texttt{ln}|x+2|\\\\:+ C}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate using a quick U-Sub]}}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-302c213f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"302c213f\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-169a2e79\" data-id=\"169a2e79\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-74f9a2b7 elementor-widget elementor-widget-toggle\" data-id=\"74f9a2b7\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1961\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1961\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 6: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1961\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1961\"><h3>Perform a partial fractions decomposition for the integrand:<\/h3>\\(\\normalsize<br \/>\\begin{align*}<br \/>\\int \\frac{1-2x}{x^2+4x^2+4x}dx<br \/>&amp;= \\int \\frac{1-2x}{x(x+2)^2}dx = \\int \\frac{A}{x}dx + \\int \\frac{B}{x+2}dx + \\int \\frac{C}{(x+2)^2}dx\\\\\\\\<br \/>&amp; 1-2x = A(x+2)^2 + Bx(x+2) + Cx<br \/>&amp; \\color{gray}{\\text{[Main equation to solve for]}} \\\\[5pt]<br \/>&amp; 1 = 4A + \\color{red}{0B} + \\color{red}{0C}<br \/>&amp; \\color{gray}{\\text{[If x = 0]}} \\\\[5pt]<br \/>&amp; \\color{blue}{A = \\frac{1}{4}}\\\\\\\\<br \/>&amp; 1-2x = A(x+2)^2 + Bx(x+2) + Cx<br \/>&amp; \\color{gray}{\\text{[Main equation to solve for]}} \\\\[5pt]<br \/>&amp; 5 = \\color{red}{0A} + \\color{red}{0B} &#8211; 2C<br \/>&amp; \\color{gray}{\\text{[If x = -2]}} \\\\[5pt]<br \/>&amp; \\color{blue}{C = -\\frac{5}{2}}\\\\\\\\<br \/>&amp; 1-2x = A(x+2)^2 + Bx(x+2) + Cx<br \/>&amp; \\color{gray}{\\text{[Main equation to solve for]}} \\\\[5pt]<br \/>&amp; -1 = 9A + 3B + C \\\\[5pt]<br \/>&amp; -1 = \\frac{9}{4} + 3B -\\frac{5}{2}<br \/>&amp; \\color{gray}{\\text{[If x = 1]}} \\\\[5pt]<br \/>&amp; \\color{blue}{B = -\\frac{1}{4}}\\\\\\\\<br \/>\\end{align*}<br \/>\\)<h3>Update the values for the partial fractions:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{1-2x}{x^3+4x^2+4x}dx<br \/>&amp; = \\int \\frac{1}{4x}dx\\:- \\int \\frac{1}{4(x+2)}dx\\:- \\int \\frac{5}{2(x+2)^2}dx<br \/>&amp; \\color{gray}{\\text{[Update the values for A and B]}} \\\\[5pt]<br \/>&amp; = \\frac{1}{2} \\left( \\int \\frac{1}{2x}dx\\:- \\frac{1}{2}\\int \\frac{1}{x+2}dx\\:- \\int \\frac{5}{(x+2)^2}dx \\right)<br \/>&amp; \\color{gray}{\\text{[Rearrange the expression]}} \\\\[5pt]<br \/>&amp; = \\frac{1}{2} \\left( \\frac{1}{2}\\texttt{ln}|x|\\:- \\frac{1}{2}\\texttt{ln}|x+2|\\:- 5\\int \\frac{1}{u^2}du \\right)<br \/>&amp; \\color{gray}{\\text{[Integrate and using quick U-Subs]}} \\\\[5pt]<br \/>&amp; = \\frac{1}{2} \\left( \\frac{1}{2}\\texttt{ln}\\left| \\frac{x}{x+2} \\right|\\:+ \\frac{5}{u} \\right) + C<br \/>&amp; \\color{gray}{\\text{[Integrate and rearrange the expressions]}} \\\\[5pt]<br \/>&amp; = \\color{green}{\\frac{1}{4}\\texttt{ln}\\left| \\frac{x}{x+2} \\right|\\:+ \\frac{5}{2(x+2)} + C}<br \/>&amp; \\color{gray}{\\text{[Simplify and substitute back to x]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 6: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Perform a partial fractions decomposition for the integrand:<\\\/h3>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{1-2x}{x^2+4x^2+4x}dx<br \\\/>&amp;= \\\\int \\\\frac{1-2x}{x(x+2)^2}dx = \\\\int \\\\frac{A}{x}dx + \\\\int \\\\frac{B}{x+2}dx + \\\\int \\\\frac{C}{(x+2)^2}dx\\\\\\\\\\\\\\\\<br \\\/>&amp; 1-2x = A(x+2)^2 + Bx(x+2) + Cx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve for]}} \\\\\\\\[5pt]<br \\\/>&amp; 1 = 4A + \\\\color{red}{0B} + \\\\color{red}{0C}<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x = 0]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{A = \\\\frac{1}{4}}\\\\\\\\\\\\\\\\<br \\\/>&amp; 1-2x = A(x+2)^2 + Bx(x+2) + Cx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve for]}} \\\\\\\\[5pt]<br \\\/>&amp; 5 = \\\\color{red}{0A} + \\\\color{red}{0B} &#8211; 2C<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x = -2]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{C = -\\\\frac{5}{2}}\\\\\\\\\\\\\\\\<br \\\/>&amp; 1-2x = A(x+2)^2 + Bx(x+2) + Cx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation to solve for]}} \\\\\\\\[5pt]<br \\\/>&amp; -1 = 9A + 3B + C \\\\\\\\[5pt]<br \\\/>&amp; -1 = \\\\frac{9}{4} + 3B -\\\\frac{5}{2}<br \\\/>&amp; \\\\color{gray}{\\\\text{[If x = 1]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{B = -\\\\frac{1}{4}}\\\\\\\\\\\\\\\\<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Update the values for the partial fractions:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{1-2x}{x^3+4x^2+4x}dx<br \\\/>&amp; = \\\\int \\\\frac{1}{4x}dx\\\\:- \\\\int \\\\frac{1}{4(x+2)}dx\\\\:- \\\\int \\\\frac{5}{2(x+2)^2}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Update the values for A and B]}} \\\\\\\\[5pt]<br \\\/>&amp; = \\\\frac{1}{2} \\\\left( \\\\int \\\\frac{1}{2x}dx\\\\:- \\\\frac{1}{2}\\\\int \\\\frac{1}{x+2}dx\\\\:- \\\\int \\\\frac{5}{(x+2)^2}dx \\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Rearrange the expression]}} \\\\\\\\[5pt]<br \\\/>&amp; = \\\\frac{1}{2} \\\\left( \\\\frac{1}{2}\\\\texttt{ln}|x|\\\\:- \\\\frac{1}{2}\\\\texttt{ln}|x+2|\\\\:- 5\\\\int \\\\frac{1}{u^2}du \\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate and using quick U-Subs]}} \\\\\\\\[5pt]<br \\\/>&amp; = \\\\frac{1}{2} \\\\left( \\\\frac{1}{2}\\\\texttt{ln}\\\\left| \\\\frac{x}{x+2} \\\\right|\\\\:+ \\\\frac{5}{u} \\\\right) + C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate and rearrange the expressions]}} \\\\\\\\[5pt]<br \\\/>&amp; = \\\\color{green}{\\\\frac{1}{4}\\\\texttt{ln}\\\\left| \\\\frac{x}{x+2} \\\\right|\\\\:+ \\\\frac{5}{2(x+2)} + C}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Simplify and substitute back to x]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e6d6c85 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e6d6c85\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-72e0b521\" data-id=\"72e0b521\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-54b119b1 elementor-widget elementor-widget-toggle\" data-id=\"54b119b1\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1421\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1421\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 7: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1421\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1421\"><h3>Perform partial fractions decomposition for the integrand:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{x-1}{x(x^2+1)}dx<br \/>&amp;= \\int \\frac{A}{x}dx\\:+ \\int \\frac{Bx+C}{x^2+1}dx<br \/>&amp; \\color{gray}{\\text{[Irreducible factor accounted for]}}<br \/>\\end{align*}<br \/>\\)<h3>Use the equating method in order to solve this equation<\/h3>\\(<br \/>\\begin{align*}<br \/>&amp;x-1 = A(x^2+1) + Bx^2 + Cx \\\\[5pt]<br \/>&amp;x-1 = Ax^2 + A +Bx^2 + Cx \\\\[5pt]<br \/>&amp;x-1 = (A+B)x^2 + Cx + A<br \/>&amp; \\color{gray}{\\text{[Equate each variable to the x variable]}} \\\\[5pt]<br \/>&amp;A+B = 0, so\\:\u00a0 A=-B \\\\[5pt]<br \/>&amp; C = 1 \\\\[5pt]<br \/>&amp;A = -1<br \/>\\end{align*}<br \/>\\)<h3>Update the new values in the partial fractions:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{x-1}{x(x^2+1)}dx<br \/>&amp;= \\int \\frac{-1}{x}dx\\:+ \\int \\frac{x+1}{x^2+1}dx<br \/>&amp; \\color{gray}{\\text{[Update the values for A and B]}} \\\\[5pt]<br \/>&amp;= -\\int \\frac{1}{x}dx\\:+ \\int \\frac{x}{x^2+1}dx\\:+ \\int \\frac{1}{x^2+1}dx<br \/>&amp; \\color{gray}{\\text{[Separate the integrals]}} \\\\[5pt]<br \/>&amp;= -\\texttt{ln}|x|\\:+ \\texttt{arctan}(x)\\:+ \\frac{1}{2}\\int \\frac{1}{u}dx<br \/>&amp; \\color{gray}{\\text{[Integrate and use a quick U-Sub]}} \\\\[5pt]<br \/>&amp;= \\color{green}{-\\texttt{ln}|x|\\:+ \\texttt{arctan}(x)\\:+ \\frac{1}{2}\\texttt{ln}(x^2+1) + C}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 7: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Perform partial fractions decomposition for the integrand:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{x-1}{x(x^2+1)}dx<br \\\/>&amp;= \\\\int \\\\frac{A}{x}dx\\\\:+ \\\\int \\\\frac{Bx+C}{x^2+1}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Irreducible factor accounted for]}}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Use the equating method in order to solve this equation<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp;x-1 = A(x^2+1) + Bx^2 + Cx \\\\\\\\[5pt]<br \\\/>&amp;x-1 = Ax^2 + A +Bx^2 + Cx \\\\\\\\[5pt]<br \\\/>&amp;x-1 = (A+B)x^2 + Cx + A<br \\\/>&amp; \\\\color{gray}{\\\\text{[Equate each variable to the x variable]}} \\\\\\\\[5pt]<br \\\/>&amp;A+B = 0, so\\\\:\\u00a0 A=-B \\\\\\\\[5pt]<br \\\/>&amp; C = 1 \\\\\\\\[5pt]<br \\\/>&amp;A = -1<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Update the new values in the partial fractions:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{x-1}{x(x^2+1)}dx<br \\\/>&amp;= \\\\int \\\\frac{-1}{x}dx\\\\:+ \\\\int \\\\frac{x+1}{x^2+1}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Update the values for A and B]}} \\\\\\\\[5pt]<br \\\/>&amp;= -\\\\int \\\\frac{1}{x}dx\\\\:+ \\\\int \\\\frac{x}{x^2+1}dx\\\\:+ \\\\int \\\\frac{1}{x^2+1}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Separate the integrals]}} \\\\\\\\[5pt]<br \\\/>&amp;= -\\\\texttt{ln}|x|\\\\:+ \\\\texttt{arctan}(x)\\\\:+ \\\\frac{1}{2}\\\\int \\\\frac{1}{u}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate and use a quick U-Sub]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{-\\\\texttt{ln}|x|\\\\:+ \\\\texttt{arctan}(x)\\\\:+ \\\\frac{1}{2}\\\\texttt{ln}(x^2+1) + C}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-303f7008 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"303f7008\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-43979e79\" data-id=\"43979e79\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7048f078 elementor-widget elementor-widget-toggle\" data-id=\"7048f078\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1881\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1881\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 8: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1881\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1881\"><h3>Rewrite the integrand and perform a partial fractions decomposition:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{x}{x^2+2x+10}dx<br \/>&amp;= \\int \\frac{x+1-1}{x^2+2x+10}dx = \\int \\frac{x+1}{x^2+2x+10}dx\\:- \\int \\frac{1}{x^2+2x+10}dx<br \/>&amp; \\color{gray}{\\text{[Sum 0 by adding 1 &#8211; 1 to the numerator]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<h3>Update the new values in the given integrands. Note that the numerator of the first integrand is half of the derivative of the denominator.<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{x}{x^2+2x+10}dx<br \/>&amp;= \\frac{1}{2} \\int \\frac{2x+2}{x^2+2x+10}dx\\:- \\int \\frac{1}{x^2+2x+10}dx<br \/>&amp; \\color{gray}{\\text{[Apply the values for A and B]}} \\\\[5pt]<br \/>&amp; = \\frac{1}{2} \\int \\frac{1}{u}du\\:- \\int \\frac{1}{(x+1)^2+9}dx<br \/>&amp; \\color{gray}{\\text{[Use a quick U-Sub and complete the square]}} \\\\[5pt]<br \/>&amp; = \\frac{1}{2} \\texttt{ln}|x^2+2x+10|\\:- \\int \\frac{1}{u^2+9}du<br \/>&amp; \\color{gray}{\\text{[Integrate and use a quick U-Sub]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<h3>Apply a trigonometric substitution. Let \\(u=3\\texttt{tan}(\\theta)\\), so \\(du=3\\texttt{sec}^2(\\theta)d\\theta\\). Updating the integrand with this substitution yields:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{1}{u^2+9}du<br \/>&amp;= \\int \\frac{1}{(\\color{red}{3\\texttt{tan}(\\theta)})^2+9}\\left( \\color{red}{3\\texttt{sec}^2(\\theta)d\\theta} \\right)<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= 3\\int \\frac{\\texttt{sec}^2(\\theta)}{9(\\texttt{tan}^2(\\theta)+1)}d\\theta = \\frac{1}{3}\\int \\frac{\\texttt{sec}^2(\\theta)}{\\texttt{sec}^2(\\theta)}d\\theta<br \/>&amp; \\color{gray}{\\text{[Rearrange the expression]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{3} \\int \\theta d\\theta = \\frac{1}{3}\\theta<br \/>&amp; \\color{gray}{\\text{[Simplify and integrate]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{3} \\texttt{arctan} \\left( \\frac{u}{3} \\right) + C<br \/>&amp; \\color{gray}{\\text{[Substitute back to u]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{3} \\texttt{arctan} \\left( \\frac{x+1}{3} \\right) + C<br \/>&amp; \\color{gray}{\\text{[Substitute back to x]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<h3>Use the previous result in the final answer<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{x}{x^2+2x+10}dx \\color{green}{= \\frac{1}{2}\\texttt{ln}|x^2+2x+10|\\:- \\frac{1}{3}\\texttt{arctan}\\left( \\frac{x+1}{3} \\right) + C}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 8: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Rewrite the integrand and perform a partial fractions decomposition:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{x}{x^2+2x+10}dx<br \\\/>&amp;= \\\\int \\\\frac{x+1-1}{x^2+2x+10}dx = \\\\int \\\\frac{x+1}{x^2+2x+10}dx\\\\:- \\\\int \\\\frac{1}{x^2+2x+10}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Sum 0 by adding 1 &#8211; 1 to the numerator]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Update the new values in the given integrands. Note that the numerator of the first integrand is half of the derivative of the denominator.<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{x}{x^2+2x+10}dx<br \\\/>&amp;= \\\\frac{1}{2} \\\\int \\\\frac{2x+2}{x^2+2x+10}dx\\\\:- \\\\int \\\\frac{1}{x^2+2x+10}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the values for A and B]}} \\\\\\\\[5pt]<br \\\/>&amp; = \\\\frac{1}{2} \\\\int \\\\frac{1}{u}du\\\\:- \\\\int \\\\frac{1}{(x+1)^2+9}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Use a quick U-Sub and complete the square]}} \\\\\\\\[5pt]<br \\\/>&amp; = \\\\frac{1}{2} \\\\texttt{ln}|x^2+2x+10|\\\\:- \\\\int \\\\frac{1}{u^2+9}du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate and use a quick U-Sub]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Apply a trigonometric substitution. Let \\\\(u=3\\\\texttt{tan}(\\\\theta)\\\\), so \\\\(du=3\\\\texttt{sec}^2(\\\\theta)d\\\\theta\\\\). Updating the integrand with this substitution yields:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{1}{u^2+9}du<br \\\/>&amp;= \\\\int \\\\frac{1}{(\\\\color{red}{3\\\\texttt{tan}(\\\\theta)})^2+9}\\\\left( \\\\color{red}{3\\\\texttt{sec}^2(\\\\theta)d\\\\theta} \\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= 3\\\\int \\\\frac{\\\\texttt{sec}^2(\\\\theta)}{9(\\\\texttt{tan}^2(\\\\theta)+1)}d\\\\theta = \\\\frac{1}{3}\\\\int \\\\frac{\\\\texttt{sec}^2(\\\\theta)}{\\\\texttt{sec}^2(\\\\theta)}d\\\\theta<br \\\/>&amp; \\\\color{gray}{\\\\text{[Rearrange the expression]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{3} \\\\int \\\\theta d\\\\theta = \\\\frac{1}{3}\\\\theta<br \\\/>&amp; \\\\color{gray}{\\\\text{[Simplify and integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{3} \\\\texttt{arctan} \\\\left( \\\\frac{u}{3} \\\\right) + C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back to u]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{3} \\\\texttt{arctan} \\\\left( \\\\frac{x+1}{3} \\\\right) + C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back to x]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Use the previous result in the final answer<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{x}{x^2+2x+10}dx \\\\color{green}{= \\\\frac{1}{2}\\\\texttt{ln}|x^2+2x+10|\\\\:- \\\\frac{1}{3}\\\\texttt{arctan}\\\\left( \\\\frac{x+1}{3} \\\\right) + C}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b94fba0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b94fba0\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6576e279\" data-id=\"6576e279\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-234d6785 elementor-widget elementor-widget-toggle\" data-id=\"234d6785\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-5921\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-5921\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 9: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-5921\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-5921\"><h3>Reexpress the intergand by performing a U-Sub. Let \\(u=\\sqrt{x+1}\\), so \\(du=\\frac{1}{2\\sqrt{x+1}}dx\\). Expanding this strategy:<\/h3>\\(\\normalsize<br \/>\\begin{align*}<br \/>u = \\sqrt{x+1} \\Rightarrow u^2 = x+1 \\Rightarrow x = u^2-1 \\\\[5pt]<br \/>du = \\frac{1}{2\\sqrt{x+1}}dx = \\frac{1}{2u}dx \\Rightarrow dx = 2udu<br \/>\\end{align*}<br \/>\\)<h3>Apply this substitutions in the integrand:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{\\color{red}{\\sqrt{x+1}}}{\\color{red}{x}} \\color{red}{dx}<br \/>&amp;= \\int \\frac{\\color{red}{u}}{\\color{red}{u^2-1}} (\\color{red}{2udu})<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= 2\\int \\frac{u^2}{u^2-1}du = 2\\int \\frac{u^2-1+1}{u^2-1}du<br \/>&amp; \\color{gray}{\\text{[Sum 0 by adding -1+1]}} \\\\[5pt]<br \/>&amp;= 2\\left[ \\int \\frac{u^2-1}{u^2-1}du\\:+ \\int \\frac{1}{u^2-1}du \\right]<br \/>&amp; \\color{gray}{\\text{[Separate the integrand]}} \\\\[5pt]<br \/>&amp;= 2\\left[ \\int du\\:+ \\int \\frac{1}{u^2-1}du \\right]<br \/>&amp; \\color{gray}{\\text{[Simplify]}} \\\\[5pt]<br \/>&amp;= 2u\\:+ 2\\int \\frac{1}{(u+1)(u-1)}du<br \/>&amp; \\color{gray}{\\text{[Apply partial fractions decomposition]}} \\\\[5pt]<br \/>&amp;= 2u\\:+ 2\\left( \\int \\frac{A}{u+1}du\\:+ \\int \\frac{B}{u-1}du \\right)\\\\\\\\\\\\<br \/>&amp; 1 = A(u-1)+B(u+1)<br \/>&amp; \\color{gray}{\\text{[Main equation]}} \\\\[5pt]<br \/>&amp; 1 = -2A+\\color{red}{0B}<br \/>&amp; \\color{gray}{\\text{[If u=-1]}} \\\\[5pt]<br \/>&amp; \\color{blue}{A=-\\frac{1}{2}}\\\\\\\\<br \/>&amp; 1 = A(u-1)+B(u+1)<br \/>&amp; \\color{gray}{\\text{[Main equation]}} \\\\[5pt]<br \/>&amp; 1 = \\color{red}{0A} + 2B<br \/>&amp; \\color{gray}{\\text{[If u=1]}} \\\\[5pt]<br \/>&amp; \\color{blue}{B=\\frac{1}{2}}\\\\\\\\<br \/>&amp; =2u + 2\\left( -\\frac{1}{2}\\int \\frac{1}{u+1}du +\\frac{1}{2}\\int \\frac{1}{u-1}du \\right)<br \/>&amp; \\color{gray}{\\text{[Update the values of A and B]}} \\\\[5pt]<br \/>&amp; = 2u +\\int \\frac{1}{u-1}du\\:- \\int \\frac{1}{u+1}du<br \/>&amp; \\color{gray}{\\text{[Simplify]}} \\\\[5pt]<br \/>&amp; = 2u + \\texttt{ln}|u-1|\\:- \\texttt{ln}|u+1| + C<br \/>&amp; \\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp; \\color{green}{ =2\\sqrt{x+1}\\:+ \\texttt{ln}\\left| \\frac{\\sqrt{x+1}-1}{\\sqrt{x+1}+1} \\right| + C}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 9: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Reexpress the intergand by performing a U-Sub. Let \\\\(u=\\\\sqrt{x+1}\\\\), so \\\\(du=\\\\frac{1}{2\\\\sqrt{x+1}}dx\\\\). Expanding this strategy:<\\\/h3>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}<br \\\/>u = \\\\sqrt{x+1} \\\\Rightarrow u^2 = x+1 \\\\Rightarrow x = u^2-1 \\\\\\\\[5pt]<br \\\/>du = \\\\frac{1}{2\\\\sqrt{x+1}}dx = \\\\frac{1}{2u}dx \\\\Rightarrow dx = 2udu<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Apply this substitutions in the integrand:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{\\\\color{red}{\\\\sqrt{x+1}}}{\\\\color{red}{x}} \\\\color{red}{dx}<br \\\/>&amp;= \\\\int \\\\frac{\\\\color{red}{u}}{\\\\color{red}{u^2-1}} (\\\\color{red}{2udu})<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= 2\\\\int \\\\frac{u^2}{u^2-1}du = 2\\\\int \\\\frac{u^2-1+1}{u^2-1}du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Sum 0 by adding -1+1]}} \\\\\\\\[5pt]<br \\\/>&amp;= 2\\\\left[ \\\\int \\\\frac{u^2-1}{u^2-1}du\\\\:+ \\\\int \\\\frac{1}{u^2-1}du \\\\right]<br \\\/>&amp; \\\\color{gray}{\\\\text{[Separate the integrand]}} \\\\\\\\[5pt]<br \\\/>&amp;= 2\\\\left[ \\\\int du\\\\:+ \\\\int \\\\frac{1}{u^2-1}du \\\\right]<br \\\/>&amp; \\\\color{gray}{\\\\text{[Simplify]}} \\\\\\\\[5pt]<br \\\/>&amp;= 2u\\\\:+ 2\\\\int \\\\frac{1}{(u+1)(u-1)}du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply partial fractions decomposition]}} \\\\\\\\[5pt]<br \\\/>&amp;= 2u\\\\:+ 2\\\\left( \\\\int \\\\frac{A}{u+1}du\\\\:+ \\\\int \\\\frac{B}{u-1}du \\\\right)\\\\\\\\\\\\\\\\\\\\\\\\<br \\\/>&amp; 1 = A(u-1)+B(u+1)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation]}} \\\\\\\\[5pt]<br \\\/>&amp; 1 = -2A+\\\\color{red}{0B}<br \\\/>&amp; \\\\color{gray}{\\\\text{[If u=-1]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{A=-\\\\frac{1}{2}}\\\\\\\\\\\\\\\\<br \\\/>&amp; 1 = A(u-1)+B(u+1)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Main equation]}} \\\\\\\\[5pt]<br \\\/>&amp; 1 = \\\\color{red}{0A} + 2B<br \\\/>&amp; \\\\color{gray}{\\\\text{[If u=1]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{blue}{B=\\\\frac{1}{2}}\\\\\\\\\\\\\\\\<br \\\/>&amp; =2u + 2\\\\left( -\\\\frac{1}{2}\\\\int \\\\frac{1}{u+1}du +\\\\frac{1}{2}\\\\int \\\\frac{1}{u-1}du \\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Update the values of A and B]}} \\\\\\\\[5pt]<br \\\/>&amp; = 2u +\\\\int \\\\frac{1}{u-1}du\\\\:- \\\\int \\\\frac{1}{u+1}du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Simplify]}} \\\\\\\\[5pt]<br \\\/>&amp; = 2u + \\\\texttt{ln}|u-1|\\\\:- \\\\texttt{ln}|u+1| + C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{ =2\\\\sqrt{x+1}\\\\:+ \\\\texttt{ln}\\\\left| \\\\frac{\\\\sqrt{x+1}-1}{\\\\sqrt{x+1}+1} \\\\right| + C}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-570abf7f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"570abf7f\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3014c0b5\" data-id=\"3014c0b5\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4004e439 elementor-widget elementor-widget-toggle\" data-id=\"4004e439\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1071\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1071\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 10: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1071\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1071\"><h3>To rearrange the integrand use a U-Sub. Let \\(u=\\sqrt{x+1}\\), so \\(du = \\frac{1}{2\\sqrt{x+1}}dx; \\Rightarrow dx = 2\\sqrt{x+1}du = 2udu\\). Find the equivalent to x in terms of u:<\/h3>\\(<br \/>\\begin{align*}<br \/>&amp; u = \\sqrt{x+1} \\\\[5pt]<br \/>&amp; u^2 = x \\:+ 1 \\\\[5pt]<br \/>&amp; x = u^2 \\:- 1<br \/>\\end{align*}<br \/>\\)<h3>Applying this substitutions in the integrand yields:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{\\sqrt{x+1}}{\\sqrt{x-1}} dx<br \/>&amp; = \\int \\frac{\\color{red}{u}}{\\sqrt{\\color{red}{(u^2-1)}-1}} \\color{red}{(2udu)} = 2 \\int \\frac{u^2}{\\sqrt{u^2-2}}du<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><p style=\"text-align: center;\"><span style=\"text-decoration: underline;\"><strong>[IN PROGRESS]<\/strong><\/span><\/p><p>\u00a0<\/p><p>\u00a0<\/p><p>\u00a0<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 10: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>To rearrange the integrand use a U-Sub. Let \\\\(u=\\\\sqrt{x+1}\\\\), so \\\\(du = \\\\frac{1}{2\\\\sqrt{x+1}}dx; \\\\Rightarrow dx = 2\\\\sqrt{x+1}du = 2udu\\\\). Find the equivalent to x in terms of u:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; u = \\\\sqrt{x+1} \\\\\\\\[5pt]<br \\\/>&amp; u^2 = x \\\\:+ 1 \\\\\\\\[5pt]<br \\\/>&amp; x = u^2 \\\\:- 1<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Applying this substitutions in the integrand yields:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{\\\\sqrt{x+1}}{\\\\sqrt{x-1}} dx<br \\\/>&amp; = \\\\int \\\\frac{\\\\color{red}{u}}{\\\\sqrt{\\\\color{red}{(u^2-1)}-1}} \\\\color{red}{(2udu)} = 2 \\\\int \\\\frac{u^2}{\\\\sqrt{u^2-2}}du<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><p style=\\\"text-align: center;\\\"><span style=\\\"text-decoration: underline;\\\"><strong>[IN PROGRESS]<\\\/strong><\\\/span><\\\/p><p>\\u00a0<\\\/p><p>\\u00a0<\\\/p><p>\\u00a0<\\\/p>\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d439768 elementor-widget elementor-widget-toggle\" data-id=\"d439768\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-2221\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2221\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 11: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2221\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2221\"><h3>Apply Weierstrass substitution. Let \\(\\texttt{sin}(\\theta)= \\frac{2x}{1+x^2}\\), \\(x=\\texttt{tan}(\\frac{\\theta}{2})\\), and \\(d\\theta=\\frac{2dx}{1+x^2}\\). Applying the substitution to the integrand yields:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{1}{1+\\color{red}{\\texttt{sin}(\\theta)}}\\color{red}{d\\theta}<br \/>&amp;= \\int \\frac{1}{1+\\color{red}{\\frac{2x}{1+x^2}}} \\left( \\color{red}{\\frac{2dx}{1+x^2}} \\right)<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= 2\\int \\left( \\frac{1}{\\frac{x^2+2x+1}{1+x^2}} \\right) \\left( \\frac{1}{1+x^2} \\right) dx<br \/>&amp; \\color{gray}{\\text{[Combine the denominator]}} \\\\[5pt]<br \/>&amp;= 2\\int \\left( \\frac{1+x^2}{(x+1)^2} \\right) \\left( \\frac{1}{1+x^2} \\right) dx<br \/>&amp; \\color{gray}{\\text{[Reverse the first fraction]}} \\\\[5pt]<br \/>&amp;= 2 \\int \\frac{1}{(x+1)^2} dx<br \/>&amp; \\color{gray}{\\text{[Simplify]}} \\\\[5pt]<br \/>&amp;= 2 \\int \\frac{1}{u^2} du = -\\frac{2}{u} + C<br \/>&amp; \\color{gray}{\\text{[Integrate using a U-Sub]}} \\\\[5pt]<br \/>&amp;= -\\frac{2}{x+1} + C<br \/>&amp; \\color{gray}{\\text{[Substitute back to x]}} \\\\[5pt]<br \/>&amp; \\color{green}{= -\\frac{2}{\\texttt{tan}(\\frac{\\theta}{2})+1} + C}<br \/>&amp; \\color{gray}{\\text{[Substitute back]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 11: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Apply Weierstrass substitution. Let \\\\(\\\\texttt{sin}(\\\\theta)= \\\\frac{2x}{1+x^2}\\\\), \\\\(x=\\\\texttt{tan}(\\\\frac{\\\\theta}{2})\\\\), and \\\\(d\\\\theta=\\\\frac{2dx}{1+x^2}\\\\). Applying the substitution to the integrand yields:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{1}{1+\\\\color{red}{\\\\texttt{sin}(\\\\theta)}}\\\\color{red}{d\\\\theta}<br \\\/>&amp;= \\\\int \\\\frac{1}{1+\\\\color{red}{\\\\frac{2x}{1+x^2}}} \\\\left( \\\\color{red}{\\\\frac{2dx}{1+x^2}} \\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= 2\\\\int \\\\left( \\\\frac{1}{\\\\frac{x^2+2x+1}{1+x^2}} \\\\right) \\\\left( \\\\frac{1}{1+x^2} \\\\right) dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Combine the denominator]}} \\\\\\\\[5pt]<br \\\/>&amp;= 2\\\\int \\\\left( \\\\frac{1+x^2}{(x+1)^2} \\\\right) \\\\left( \\\\frac{1}{1+x^2} \\\\right) dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Reverse the first fraction]}} \\\\\\\\[5pt]<br \\\/>&amp;= 2 \\\\int \\\\frac{1}{(x+1)^2} dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Simplify]}} \\\\\\\\[5pt]<br \\\/>&amp;= 2 \\\\int \\\\frac{1}{u^2} du = -\\\\frac{2}{u} + C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate using a U-Sub]}} \\\\\\\\[5pt]<br \\\/>&amp;= -\\\\frac{2}{x+1} + C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back to x]}} \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{= -\\\\frac{2}{\\\\texttt{tan}(\\\\frac{\\\\theta}{2})+1} + C}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-bd5dc66 elementor-widget elementor-widget-toggle\" data-id=\"bd5dc66\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1981\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1981\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 12: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1981\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1981\"><h3>Apply Weierstrass substitution. Let \\(\\texttt{sin}(\\theta)=\\frac{2x}{1+x^2}\\), \\(\\texttt{cos}(\\theta)=\\frac{1-x^2}{1+x^2}\\), \\(x=\\texttt{tan}(\\frac{\\theta}{2})\\), and \\(d\\theta = \\frac{2dx}{1+x^2}\\). Applying this substitution in the integrand leads to:<\/h3>\n\\(\n\\begin{align*}\n\\int \\frac{1}{1+\\color{red}{\\texttt{sin}(\\theta)} + \\color{red}{\\texttt{cos}(\\theta)}} \\color{red}{d\\theta}\n&amp;= \\int \\frac{1}{1+\\color{red}{\\frac{2x}{1+x^2}} + \\color{red}{\\frac{1-x^2}{1+x^2}}} \\color{red}{\\left( \\frac{2}{1+x^2}dx \\right)}\n&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]\n&amp;= 2 \\int \\left( \\frac{1}{\\frac{1+x^2+2x+1-x^2}{1+x^2}} \\right) \\left( \\frac{1}{1+x^2} \\right) dx\n&amp; \\color{gray}{\\text{[Merge the denominator]}} \\\\[5pt]\n&amp;= 2 \\int \\left( \\frac{1}{\\frac{2x+2}{1+x^2}} \\right) \\left( \\frac{1}{1+x^2} \\right) dx\n&amp; \\color{gray}{\\text{[Simplify]}} \\\\[5pt]\n&amp;= 2 \\int \\left( \\frac{1+x^2}{2x+2} \\right) \\left( \\frac{1}{1+x^2} \\right) dx\n= 2 \\int \\frac{1}{2x+2} dx = \\int \\frac{1}{x+1}dx \\\\[5pt]\n&amp;= \\texttt{ln}|x+1| + C\n&amp; \\color{gray}{\\text{[Integrate]}} \\\\[5pt]\n&amp; \\color{green}{= \\texttt{ln}\\left| \\texttt{tan} \\left( \\frac{\\theta}{2} \\right) +1 \\right| + C}\n&amp; \\color{gray}{\\text{[Substitute back]}}\n\\end{align*}\n\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 12: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Apply Weierstrass substitution. Let \\\\(\\\\texttt{sin}(\\\\theta)=\\\\frac{2x}{1+x^2}\\\\), \\\\(\\\\texttt{cos}(\\\\theta)=\\\\frac{1-x^2}{1+x^2}\\\\), \\\\(x=\\\\texttt{tan}(\\\\frac{\\\\theta}{2})\\\\), and \\\\(d\\\\theta = \\\\frac{2dx}{1+x^2}\\\\). Applying this substitution in the integrand leads to:<\\\/h3>\\n\\\\(\\n\\\\begin{align*}\\n\\\\int \\\\frac{1}{1+\\\\color{red}{\\\\texttt{sin}(\\\\theta)} + \\\\color{red}{\\\\texttt{cos}(\\\\theta)}} \\\\color{red}{d\\\\theta}\\n&amp;= \\\\int \\\\frac{1}{1+\\\\color{red}{\\\\frac{2x}{1+x^2}} + \\\\color{red}{\\\\frac{1-x^2}{1+x^2}}} \\\\color{red}{\\\\left( \\\\frac{2}{1+x^2}dx \\\\right)}\\n&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]\\n&amp;= 2 \\\\int \\\\left( \\\\frac{1}{\\\\frac{1+x^2+2x+1-x^2}{1+x^2}} \\\\right) \\\\left( \\\\frac{1}{1+x^2} \\\\right) dx\\n&amp; \\\\color{gray}{\\\\text{[Merge the denominator]}} \\\\\\\\[5pt]\\n&amp;= 2 \\\\int \\\\left( \\\\frac{1}{\\\\frac{2x+2}{1+x^2}} \\\\right) \\\\left( \\\\frac{1}{1+x^2} \\\\right) dx\\n&amp; \\\\color{gray}{\\\\text{[Simplify]}} \\\\\\\\[5pt]\\n&amp;= 2 \\\\int \\\\left( \\\\frac{1+x^2}{2x+2} \\\\right) \\\\left( \\\\frac{1}{1+x^2} \\\\right) dx\\n= 2 \\\\int \\\\frac{1}{2x+2} dx = \\\\int \\\\frac{1}{x+1}dx \\\\\\\\[5pt]\\n&amp;= \\\\texttt{ln}|x+1| + C\\n&amp; \\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]\\n&amp; \\\\color{green}{= \\\\texttt{ln}\\\\left| \\\\texttt{tan} \\\\left( \\\\frac{\\\\theta}{2} \\\\right) +1 \\\\right| + C}\\n&amp; \\\\color{gray}{\\\\text{[Substitute back]}}\\n\\\\end{align*}\\n\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-28d409f4 elementor-widget elementor-widget-toggle\" data-id=\"28d409f4\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-6841\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-6841\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 13: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-6841\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-6841\"><h3>Apply Weierstrass substitution. Let \\(\\texttt{sin}(\\theta)=\\frac{2x}{1+x^2}\\), \\(\\texttt{cos}(\\theta)=\\frac{1-x^2}{1+x^2}\\), \\(x=\\texttt{tan}(\\frac{\\theta}{2})\\), and \\(d\\theta = \\frac{2dx}{1+x^2}\\). Applying this substitution in the integrand leads to:<\/h3>\\(<br \/>\\begin{align*}<br \/>\\int \\frac{\\color{red}{\\texttt{sin}(\\theta)}}{\\color{red}{\\texttt{sin}(\\theta)} + \\color{red}{\\texttt{cos}(\\theta)}} \\color{red}{d\\theta}<br \/>&amp;= \\int \\frac{\\color{red}{\\frac{2x}{1+x^2}}}{\\color{red}{\\frac{2x}{1+x^2}} + \\color{red}{\\frac{1-x^2}{1+x^2}}} \\color{red}{\\left( \\frac{2}{1+x^2}dx \\right)}<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= 4 \\int \\frac{\\frac{x}{(1+x^2)^2}}{\\frac{-x^2+2x+1}{1+x^2}}dx<br \/>&amp; \\color{gray}{\\text{[Manipulate algebraically]}} \\\\[5pt]<br \/>&amp;= 4 \\int \\left( \\frac{x}{(1+x^2)^2} \\right) \\left( \\frac{1+x^2}{-x^2+2x+1} \\right) dx<br \/>&amp; \\color{gray}{\\text{[Reverse the denominator]}} \\\\[5pt]<br \/>&amp;= -4 \\int \\frac{x}{(1+x^2)(x^2-2x-1)}dx<br \/>&amp; \\color{gray}{\\text{[Simplify]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<h3>In order to apply partial fractions decomposition use the quadratic formula for the second factor in the denominator:<\/h3>\\(<br \/>\\begin{align*}<br \/>&amp; x = \\frac{-b \\pm \\sqrt{b^2-4ac}}{2a} = \\frac{-(-2) \\pm \\sqrt{(-2)^2-4(1)(-1)}}{2(1)} = \\frac{2 \\pm \\sqrt{4+4}}{2} = \\frac{2 \\pm \\sqrt{8}}{2} = \\frac{2 \\pm 2\\sqrt{2}}{2} = 1 \\pm \\sqrt{2} \\\\[5pt]<br \/>&amp; x_1 = 1 + \\sqrt{2} \\\\[5pt]<br \/>&amp; x_2 = 1 &#8211; \\sqrt{2} \\\\[5pt]<br \/>&amp; x^2-2x-1 = (x &#8211; 1 + \\sqrt{2})(x &#8211; 1 &#8211; \\sqrt{2})<br \/>\\end{align*}<br \/>\\)<h3>Update the factor in the denominator with the new factorization result to apply partial fractions decomposition:<\/h3>\\(<br \/>\\begin{align*}<br \/>&amp;-4\\int \\frac{x}{(1+x^2)\\color{red}{(x^2-2x-1)}}dx = -4\\int \\frac{x}{(1+x^2)\\color{red}{(x-1+\\sqrt{2})(x-1-\\sqrt{2})}}dx \\\\[5pt]<br \/>&amp;= -4\\left( \\int \\frac{Ax+B}{1+x^2}dx + \\int \\frac{C}{x-1+\\sqrt{2}}dx + \\int \\frac{D}{x-1-\\sqrt{2}}dx \\right) \\\\\\\\<br \/>&amp; \\color{gray}{\\text{[Use the equating method]}} \\\\[5pt]<br \/>&amp;x = (Ax+B)(x-1+\\sqrt{2})(x-1-\\sqrt{2}) + \\color{blue}{C(x^2+1)(x-1-\\sqrt{2})} + \\color{red}{D(x^2+1)(x-1+\\sqrt{2})} \\\\[5pt]<br \/>&amp; x = Ax^3-2Ax^2-Ax+Bx^2-2Bx-B + \\color{blue}{Cx^3-Cx^2-\\sqrt{2}Cx^2+Cx-C-\\sqrt{2}C} + \\color{red}{Dx^3-Dx^2+\\sqrt{2}Dx^2+Dx-D+\\sqrt{2}D} \\\\[5pt]<br \/>&amp; x = \\color{red}{[A+C+D]}x^3 + \\color{blue}{[-2A+B-C(1+\\sqrt{2})-D(1-\\sqrt{2})]}x^2 + \\color{green}{[-A-2B+C + D]}x + \\color{orange}{[-B-C(1+\\sqrt{2})-D(1-\\sqrt{2})]} \\\\\\\\<br \/>&amp; \\color{gray}{\\text{[Solve the resulting equation system using any method]}} \\\\[5pt]<br \/>&amp;A+C+D=0 \\\\[5pt]<br \/>&amp;-2A+B-C-\\sqrt{2}C-D+\\sqrt{2}D=0 \\\\[5pt]<br \/>&amp;-A-2B+C+D=1 \\\\[5pt]<br \/>&amp;-B-C-\\sqrt{2}C-D+\\sqrt{2}D=0 \\\\[5pt]<br \/>&amp; A = -\\frac{1}{4}\\\\[5pt]<br \/>&amp; B = -\\frac{1}{4} \\\\[5pt]<br \/>&amp; C = \\frac{1}{8} \\\\[5pt]<br \/>&amp; D = \\frac{1}{8} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<h3>Update the newfound values into the partial fraction integrals:<\/h3>\\(<br \/>\\begin{align*}<br \/>-4&amp;\\int \\frac{x}{(1+x^2)(x-1+\\sqrt{2})(x-1-\\sqrt{2})}dx\\\\<br \/>&amp;= -4\\left( \\int \\frac{-\\frac{1}{4}x-\\frac{1}{4}}{1+x^2}dx + \\int \\frac{\\frac{1}{8}}{x-1+\\sqrt{2}}dx + \\int \\frac{\\frac{1}{8}}{x-1-\\sqrt{2}}\\right)<br \/>&amp; \\color{gray}{\\text{[Update]}} \\\\[5pt]<br \/>&amp;= \\int \\frac{x+1}{1+x^2}dx\\:- \\frac{1}{2}\\int \\frac{1}{x-1+\\sqrt{2}}dx\\:- \\frac{1}{2}\\int \\frac{1}{x-1-\\sqrt{2}}dx \\\\[5pt]<br \/>&amp;= \\int \\frac{x}{1+x^2}dx + \\int \\frac{1}{1+x^2}dx\\: &#8211; \\frac{1}{2} \\left( \\texttt{ln}|x-1+\\sqrt{2}|\\:+ \\texttt{Ln}|x-1-\\sqrt{2}| \\right) + C<br \/>&amp; \\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp; = \\frac{1}{2}\\int \\frac{1}{u}du + \\int \\frac{1}{1+x^2}dx\\: &#8211; \\frac{1}{2} \\texttt{ln}|(x-1+\\sqrt{2})(x-1-\\sqrt{2})| + C \\\\[5pt]<br \/>&amp; = \\frac{1}{2} \\texttt{ln}|u| + \\texttt{arctan}(x)\\:- \\frac{1}{2} \\texttt{ln}|x^2-2x-1| + C \\\\[5pt]<br \/>&amp; = \\frac{1}{2} \\texttt{ln}(x^2+1)\\:- \\frac{1}{2} \\texttt{ln}|x^2-2x-1| + \\texttt{arctan}(x) + C \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\texttt{ln} \\left| \\frac{x^2+1}{x^2-2x-1} \\right| +\\texttt{arctan}(x) + C<br \/>&amp; \\color{gray}{\\text{[Substitute back]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{1}{2} \\texttt{ln} \\left| \\frac{\\texttt{tan}^2(\\frac{\\theta}{2})+1}{\\texttt{tan}^2(\\theta)-2\\texttt{tan}(\\frac{\\theta}{2})-1} \\right| + \\texttt{arctan} \\left( \\texttt{tan} \\left( \\frac{\\theta}{2} \\right) \\right) + C}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 13: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Apply Weierstrass substitution. Let \\\\(\\\\texttt{sin}(\\\\theta)=\\\\frac{2x}{1+x^2}\\\\), \\\\(\\\\texttt{cos}(\\\\theta)=\\\\frac{1-x^2}{1+x^2}\\\\), \\\\(x=\\\\texttt{tan}(\\\\frac{\\\\theta}{2})\\\\), and \\\\(d\\\\theta = \\\\frac{2dx}{1+x^2}\\\\). Applying this substitution in the integrand leads to:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\frac{\\\\color{red}{\\\\texttt{sin}(\\\\theta)}}{\\\\color{red}{\\\\texttt{sin}(\\\\theta)} + \\\\color{red}{\\\\texttt{cos}(\\\\theta)}} \\\\color{red}{d\\\\theta}<br \\\/>&amp;= \\\\int \\\\frac{\\\\color{red}{\\\\frac{2x}{1+x^2}}}{\\\\color{red}{\\\\frac{2x}{1+x^2}} + \\\\color{red}{\\\\frac{1-x^2}{1+x^2}}} \\\\color{red}{\\\\left( \\\\frac{2}{1+x^2}dx \\\\right)}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= 4 \\\\int \\\\frac{\\\\frac{x}{(1+x^2)^2}}{\\\\frac{-x^2+2x+1}{1+x^2}}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Manipulate algebraically]}} \\\\\\\\[5pt]<br \\\/>&amp;= 4 \\\\int \\\\left( \\\\frac{x}{(1+x^2)^2} \\\\right) \\\\left( \\\\frac{1+x^2}{-x^2+2x+1} \\\\right) dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Reverse the denominator]}} \\\\\\\\[5pt]<br \\\/>&amp;= -4 \\\\int \\\\frac{x}{(1+x^2)(x^2-2x-1)}dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Simplify]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>In order to apply partial fractions decomposition use the quadratic formula for the second factor in the denominator:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; x = \\\\frac{-b \\\\pm \\\\sqrt{b^2-4ac}}{2a} = \\\\frac{-(-2) \\\\pm \\\\sqrt{(-2)^2-4(1)(-1)}}{2(1)} = \\\\frac{2 \\\\pm \\\\sqrt{4+4}}{2} = \\\\frac{2 \\\\pm \\\\sqrt{8}}{2} = \\\\frac{2 \\\\pm 2\\\\sqrt{2}}{2} = 1 \\\\pm \\\\sqrt{2} \\\\\\\\[5pt]<br \\\/>&amp; x_1 = 1 + \\\\sqrt{2} \\\\\\\\[5pt]<br \\\/>&amp; x_2 = 1 &#8211; \\\\sqrt{2} \\\\\\\\[5pt]<br \\\/>&amp; x^2-2x-1 = (x &#8211; 1 + \\\\sqrt{2})(x &#8211; 1 &#8211; \\\\sqrt{2})<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Update the factor in the denominator with the new factorization result to apply partial fractions decomposition:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp;-4\\\\int \\\\frac{x}{(1+x^2)\\\\color{red}{(x^2-2x-1)}}dx = -4\\\\int \\\\frac{x}{(1+x^2)\\\\color{red}{(x-1+\\\\sqrt{2})(x-1-\\\\sqrt{2})}}dx \\\\\\\\[5pt]<br \\\/>&amp;= -4\\\\left( \\\\int \\\\frac{Ax+B}{1+x^2}dx + \\\\int \\\\frac{C}{x-1+\\\\sqrt{2}}dx + \\\\int \\\\frac{D}{x-1-\\\\sqrt{2}}dx \\\\right) \\\\\\\\\\\\\\\\<br \\\/>&amp; \\\\color{gray}{\\\\text{[Use the equating method]}} \\\\\\\\[5pt]<br \\\/>&amp;x = (Ax+B)(x-1+\\\\sqrt{2})(x-1-\\\\sqrt{2}) + \\\\color{blue}{C(x^2+1)(x-1-\\\\sqrt{2})} + \\\\color{red}{D(x^2+1)(x-1+\\\\sqrt{2})} \\\\\\\\[5pt]<br \\\/>&amp; x = Ax^3-2Ax^2-Ax+Bx^2-2Bx-B + \\\\color{blue}{Cx^3-Cx^2-\\\\sqrt{2}Cx^2+Cx-C-\\\\sqrt{2}C} + \\\\color{red}{Dx^3-Dx^2+\\\\sqrt{2}Dx^2+Dx-D+\\\\sqrt{2}D} \\\\\\\\[5pt]<br \\\/>&amp; x = \\\\color{red}{[A+C+D]}x^3 + \\\\color{blue}{[-2A+B-C(1+\\\\sqrt{2})-D(1-\\\\sqrt{2})]}x^2 + \\\\color{green}{[-A-2B+C + D]}x + \\\\color{orange}{[-B-C(1+\\\\sqrt{2})-D(1-\\\\sqrt{2})]} \\\\\\\\\\\\\\\\<br \\\/>&amp; \\\\color{gray}{\\\\text{[Solve the resulting equation system using any method]}} \\\\\\\\[5pt]<br \\\/>&amp;A+C+D=0 \\\\\\\\[5pt]<br \\\/>&amp;-2A+B-C-\\\\sqrt{2}C-D+\\\\sqrt{2}D=0 \\\\\\\\[5pt]<br \\\/>&amp;-A-2B+C+D=1 \\\\\\\\[5pt]<br \\\/>&amp;-B-C-\\\\sqrt{2}C-D+\\\\sqrt{2}D=0 \\\\\\\\[5pt]<br \\\/>&amp; A = -\\\\frac{1}{4}\\\\\\\\[5pt]<br \\\/>&amp; B = -\\\\frac{1}{4} \\\\\\\\[5pt]<br \\\/>&amp; C = \\\\frac{1}{8} \\\\\\\\[5pt]<br \\\/>&amp; D = \\\\frac{1}{8} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h3>Update the newfound values into the partial fraction integrals:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>-4&amp;\\\\int \\\\frac{x}{(1+x^2)(x-1+\\\\sqrt{2})(x-1-\\\\sqrt{2})}dx\\\\\\\\<br \\\/>&amp;= -4\\\\left( \\\\int \\\\frac{-\\\\frac{1}{4}x-\\\\frac{1}{4}}{1+x^2}dx + \\\\int \\\\frac{\\\\frac{1}{8}}{x-1+\\\\sqrt{2}}dx + \\\\int \\\\frac{\\\\frac{1}{8}}{x-1-\\\\sqrt{2}}\\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Update]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\int \\\\frac{x+1}{1+x^2}dx\\\\:- \\\\frac{1}{2}\\\\int \\\\frac{1}{x-1+\\\\sqrt{2}}dx\\\\:- \\\\frac{1}{2}\\\\int \\\\frac{1}{x-1-\\\\sqrt{2}}dx \\\\\\\\[5pt]<br \\\/>&amp;= \\\\int \\\\frac{x}{1+x^2}dx + \\\\int \\\\frac{1}{1+x^2}dx\\\\: &#8211; \\\\frac{1}{2} \\\\left( \\\\texttt{ln}|x-1+\\\\sqrt{2}|\\\\:+ \\\\texttt{Ln}|x-1-\\\\sqrt{2}| \\\\right) + C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp; = \\\\frac{1}{2}\\\\int \\\\frac{1}{u}du + \\\\int \\\\frac{1}{1+x^2}dx\\\\: &#8211; \\\\frac{1}{2} \\\\texttt{ln}|(x-1+\\\\sqrt{2})(x-1-\\\\sqrt{2})| + C \\\\\\\\[5pt]<br \\\/>&amp; = \\\\frac{1}{2} \\\\texttt{ln}|u| + \\\\texttt{arctan}(x)\\\\:- \\\\frac{1}{2} \\\\texttt{ln}|x^2-2x-1| + C \\\\\\\\[5pt]<br \\\/>&amp; = \\\\frac{1}{2} \\\\texttt{ln}(x^2+1)\\\\:- \\\\frac{1}{2} \\\\texttt{ln}|x^2-2x-1| + \\\\texttt{arctan}(x) + C \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\texttt{ln} \\\\left| \\\\frac{x^2+1}{x^2-2x-1} \\\\right| +\\\\texttt{arctan}(x) + C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{1}{2} \\\\texttt{ln} \\\\left| \\\\frac{\\\\texttt{tan}^2(\\\\frac{\\\\theta}{2})+1}{\\\\texttt{tan}^2(\\\\theta)-2\\\\texttt{tan}(\\\\frac{\\\\theta}{2})-1} \\\\right| + \\\\texttt{arctan} \\\\left( \\\\texttt{tan} \\\\left( \\\\frac{\\\\theta}{2} \\\\right) \\\\right) + C}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-97d0f71 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"97d0f71\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-436084c\" data-id=\"436084c\" 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href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/fundamentals-differentiation\/\" class=\"elementor-sub-item\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11537\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-formulas-update\/\" class=\"elementor-sub-item\">Differentiation Formulas<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11409\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-techniques\/\" class=\"elementor-sub-item\">Differentiation Techniques<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-11411\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-applications\/\" class=\"elementor-sub-item\">Differentiation Applications<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-15049\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/lhopitals-rule\/\" class=\"elementor-sub-item\">L\u2019H\u00f4pital\u2019s Rule<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10229\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-applications\/curve-analysis\/\" class=\"elementor-sub-item\">Curve Analysis<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11435\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-applications\/related-rates\/\" class=\"elementor-sub-item\">Related Rates<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20868\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/problem-set-differentiation\/\" class=\"elementor-sub-item\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14329\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Integration<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-14551\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/fundamentals-integration\/\" class=\"elementor-sub-item\">Fundamentals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10226\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/riemann-sums\/\" class=\"elementor-sub-item\">Riemann Sums<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11365\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/fundamental-theorem-of-calculus\/\" class=\"elementor-sub-item\">Fundamental Theorem of Calculus<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11530\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-formulas-update\/\" class=\"elementor-sub-item\">Integration Formulas<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-9815\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-techinques\/\" class=\"elementor-sub-item\">Integration Techinques<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-13700\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-applications\/\" class=\"elementor-sub-item\">Integration Applications<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19550\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/\" class=\"elementor-sub-item\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14330\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Sequences and Series<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11364\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/\" class=\"elementor-sub-item\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-9804\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/convergence-test\/\" class=\"elementor-sub-item\">Convergence Test<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10233\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/power-series\/\" class=\"elementor-sub-item\">Power Series<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11363\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/taylor-series\/\" class=\"elementor-sub-item\">Taylor Series<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-21607\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/problem-set-sequences-series\/\" class=\"elementor-sub-item\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14331\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Parametric Equations<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12174\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/fundamentals-of-parametric-equations\/\" class=\"elementor-sub-item\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19376\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/polar-coordinates-calculus\/\" class=\"elementor-sub-item\">Polar Coordinates<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11366\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/precalculus-resources\/unit-circle\/\" class=\"elementor-sub-item\">Unit Circle<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12198\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/calculus-parametric-equation\/\" class=\"elementor-sub-item\">Calculus of Parametric Equations<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-21606\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/problem-set-parametric-eqs-2\/\" class=\"elementor-sub-item\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12615\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/virtual-lessons-parametric-equations\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10232\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/ap-calculus-frqs\/\" class=\"elementor-item\">AP Calculus Prep<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t<div class=\"elementor-menu-toggle\" role=\"button\" tabindex=\"0\" aria-label=\"Menu Toggle\" aria-expanded=\"false\">\n\t\t\t<i aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--open eicon-menu-bar\"><\/i><i aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--close eicon-close\"><\/i>\t\t<\/div>\n\t\t\t\t\t<nav class=\"elementor-nav-menu--dropdown elementor-nav-menu__container\" aria-hidden=\"true\">\n\t\t\t\t<ul id=\"menu-2-7f5d6b4\" class=\"elementor-nav-menu\"><li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-14333\"><a href=\"https:\/\/onlinemathtutor.co\/index.php\/course-content\/calculus-resources\/\" class=\"elementor-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2021\/06\/Home-Icon-Blue.png\"  width=\"20\"><\/a><\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14328\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Limits<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12209\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/fundamentals-limits\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-14279\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/limit-formulas\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Limit Formulas<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11442\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/continuity\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Continuity<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-11426\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/indeterminate-limits\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Indeterminate Forms<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12217\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/lhopitals-rule\/\" class=\"elementor-sub-item\" tabindex=\"-1\">L\u2019H\u00f4pital\u2019s Rule<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12629\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/virtual-lessons-limits\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14327\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Differentiation<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-14856\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/fundamentals-differentiation\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11537\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-formulas-update\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Differentiation Formulas<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11409\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-techniques\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Differentiation Techniques<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-11411\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-applications\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Differentiation Applications<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-15049\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/lhopitals-rule\/\" class=\"elementor-sub-item\" tabindex=\"-1\">L\u2019H\u00f4pital\u2019s Rule<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10229\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-applications\/curve-analysis\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Curve Analysis<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11435\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-applications\/related-rates\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Related Rates<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20868\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/problem-set-differentiation\/\" class=\"elementor-sub-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14329\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Integration<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-14551\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/fundamentals-integration\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamentals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10226\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/riemann-sums\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Riemann Sums<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11365\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/fundamental-theorem-of-calculus\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamental Theorem of Calculus<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11530\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-formulas-update\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Integration Formulas<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-9815\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-techinques\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Integration Techinques<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-13700\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-applications\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Integration Applications<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19550\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/\" class=\"elementor-sub-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14330\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Sequences and Series<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11364\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-9804\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/convergence-test\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Convergence Test<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10233\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/power-series\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Power Series<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11363\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/taylor-series\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Taylor Series<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-21607\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/problem-set-sequences-series\/\" class=\"elementor-sub-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14331\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Parametric Equations<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12174\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/fundamentals-of-parametric-equations\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19376\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/polar-coordinates-calculus\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Polar Coordinates<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11366\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/precalculus-resources\/unit-circle\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Unit Circle<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12198\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/calculus-parametric-equation\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Calculus of Parametric Equations<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-21606\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/problem-set-parametric-eqs-2\/\" class=\"elementor-sub-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12615\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/virtual-lessons-parametric-equations\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10232\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/ap-calculus-frqs\/\" class=\"elementor-item\" tabindex=\"-1\">AP Calculus Prep<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-1c1e1cd\" data-id=\"1c1e1cd\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-788e7a3 elementor-widget elementor-widget-spacer\" data-id=\"788e7a3\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Practice Problems: Partial Fractions Although partial fraction decomposition (or simply partial fractions) is an algebraic, not an integration technique, it is useful for integrating rational functions. When decomposing the ration function, remember to echo any repeated factors. 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