{"id":20092,"date":"2022-02-23T02:49:09","date_gmt":"2022-02-23T02:49:09","guid":{"rendered":"https:\/\/onlinemathtutor.co\/?page_id=20092"},"modified":"2022-05-19T21:02:12","modified_gmt":"2022-05-19T21:02:12","slug":"integration-practice-problems-simpsons","status":"publish","type":"page","link":"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-simpsons\/","title":{"rendered":"Integration Practice Problems &#8211; [Simpsons]"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"20092\" class=\"elementor elementor-20092\" 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: Simpson's Rule (1\/3)<\/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: 18px;\">Also known as Simpson&#8217;s \\(\\textstyle\\frac{1}{3}\\) Rule is a numerical integration technique that improves upon the Trapezoidal Rule by utilizing the geometry of parabolic arcs. The number of partitions \\(n\\) must be even.<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 18px;\">\\(\\displaystyle S_n=\\frac{b-a}{3n}\\big( f(x_0)+4f(x_1)+2f(x_2)+\\cdots+4f(x_{n-1})+2f(x_n) \\big)\\)<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 18px;\">\\(\\displaystyle \\Delta x=\\frac{b-a}{n}\\quad\n\\displaystyle x_k=a+k\\Delta x\\)<\/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-simpsons\/?playlist=b24a5c0&#038;video=1adcc07\" 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-simpsons\/?playlist=b24a5c0&#038;video=1adcc07\" >\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;\">Approximate the given interval using <strong>Simpson Rule<\/strong> using the indicated partitions \\(n\\).\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-546685f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"546685f\" 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-be1922d\" data-id=\"be1922d\" 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-cfcf2c6 elementor-widget elementor-widget-text-editor\" data-id=\"cfcf2c6\" 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\">\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int (2x+1)^2dx\\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int x\\sqrt{2x+3}dx\\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{x}{2x+1}dx\\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\frac{x^3}{x^2-4}dx \\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int \\texttt{e}^{x^2+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-6a89fe3\" data-id=\"6a89fe3\" 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-79c5eff elementor-widget elementor-widget-text-editor\" data-id=\"79c5eff\" 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{\\texttt{e}^{2x}}{1+\\texttt{e}^{2x}}} dx \\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int_1^\\texttt{e}\u00a0{\\frac{\\texttt{ln}(x)}{x} dx} \\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int {\\frac{\\texttt{sin}(x)}{1+\\texttt{cos}(x)}} dx \\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int {\\frac{\\texttt{e}^x}{1+\\texttt{e}^{2x}}} dx \\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int {\\frac{1}{\\texttt{e}^x + \\texttt{e}^{-x}}} 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-6a938fe\" data-id=\"6a938fe\" 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-c5a479a elementor-widget elementor-widget-text-editor\" data-id=\"c5a479a\" 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\">\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int {\\frac{1}{\\sqrt{x}(x+1)}} dx \\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int {\\frac{x}{\\sqrt{1-x^4}}} dx \\)<\/li>\n \t<li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\int {\\frac{x^2}{x^2+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\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-96a1c60 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"96a1c60\" 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-83bd0b7\" data-id=\"83bd0b7\" 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-e0fe46a elementor-align-left elementor-icon-list--layout-traditional elementor-list-item-link-full_width elementor-widget elementor-widget-icon-list\" data-id=\"e0fe46a\" 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-a9e2d37 elementor-widget elementor-widget-spacer\" data-id=\"a9e2d37\" 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-52bdde0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"52bdde0\" 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-a6845f0\" data-id=\"a6845f0\" 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-95508a8 elementor-widget elementor-widget-toggle\" data-id=\"95508a8\" 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-1561\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1561\" 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-1561\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1561\"><p><span style=\"font-size: 14pt;\">By setting \\(u=2x+1\\) gives \\(du=2dx\\) and substituting these values into the integral leads to the following:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int (2x+1)^2 dx<br \/>&amp;= \\int ( \\color{red}{\\underbrace{\\color{black}{2x+1}}_{\\large u}} )^2 \\color{red}{\\underbrace{\\color{black}{dx}}_{\\large \\frac{du}{2}}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2}\\int u^2 <br \/>&amp;\\color{gray}{\\text{[Apply the Substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2}\\cdot\\frac{u^3}{3} + C<br \/>&amp;\\color{gray}{\\text{[Apply the Power Rule]}}\\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{(2x+1)^3}{6} + C}<br \/>&amp;\\color{gray}{\\text{[Convert back to } x \\text{]}}<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\":\"<p><span style=\\\"font-size: 14pt;\\\">By setting \\\\(u=2x+1\\\\) gives \\\\(du=2dx\\\\) and substituting these values into the integral leads to the following:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int (2x+1)^2 dx<br \\\/>&amp;= \\\\int ( \\\\color{red}{\\\\underbrace{\\\\color{black}{2x+1}}_{\\\\large u}} )^2 \\\\color{red}{\\\\underbrace{\\\\color{black}{dx}}_{\\\\large \\\\frac{du}{2}}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2}\\\\int u^2 <br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the Substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2}\\\\cdot\\\\frac{u^3}{3} + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the Power Rule]}}\\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{(2x+1)^3}{6} + C}<br \\\/>&amp;\\\\color{gray}{\\\\text{[Convert back to } x \\\\text{]}}<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-08eff87 elementor-widget elementor-widget-toggle\" data-id=\"08eff87\" 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-9371\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-9371\" 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-9371\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-9371\"><p><span style=\"font-size: 18.6667px;\">Let \\(u=2x+3\\), \\(x = \\frac{u-3}{2}\\), and \\(du=2dx\\). Substituting these values into the integral leads to the following:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int x\\sqrt{2x+3} dx<br \/>&amp;= \\int \\color{red}{\\underbrace{\\color{black}{x}}_{\\large \\frac{u-3}{2}}} \\color{red}{\\underbrace{\\color{black}{\\sqrt{2x+3}}}_{\\large u}}<br \/>\\color{red}{\\underbrace{\\color{black}{dx}}_{\\large \\frac{du}{2}}} \\\\[5pt]<br \/>&amp;= \\frac{1}{4}\\int (u-3)\\sqrt{u} du <br \/>&amp;\\color{gray}{\\text{[Apply the Substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{4} \\left(\\int{u\\sqrt{u}\\: du}\\: -\\: 3\\int \\sqrt{u} \\: du\\right)<br \/>&amp;\\color{gray}{\\text{[Distribute the Integrals]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{4} \\left(\\int{u^\\frac{3}{2}\\: du}\\: -\\: 3\\int u^\\frac{1}{2} \\: du\\right)<br \/>&amp;\\color{gray}{\\text{[Use power rules to simplify]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{4}\\left(\\frac{u^\\frac{5}{2}}{\\frac{5}{2}} &#8211; \\frac{u^\\frac{3}{2}}{\\frac{3}{2}}\u00a0\\right) + C<br \/>&amp;\\color{gray}{\\text{[Apply the Power Rule]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{4}\\left[\\left(\\frac{2}{5}\\right)u^\\frac{5}{2} &#8211; \\left(\\frac{2}{3}\\right)u^\\frac{3}{2}\u00a0\\right] + C<br \/>&amp;\\color{gray}{\\text{[Simplify the fractions]}} \\\\[5pt]<br \/>&amp; = \\frac{1}{10}u^\\frac{5}{2}\\: &#8211; \\frac{1}{6}u^\\frac{3}{2} + C<br \/>&amp;\\color{gray}{\\text{[Distribute the outside fraction]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{1}{10}(2x+3)^\\frac{5}{2}\\: &#8211; \\frac{1}{6}(2x+3)^\\frac{3}{2} + C}<br \/>&amp;\\color{gray}{\\text{[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 2: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Let \\\\(u=2x+3\\\\), \\\\(x = \\\\frac{u-3}{2}\\\\), and \\\\(du=2dx\\\\). Substituting these values into the integral leads to the following:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int x\\\\sqrt{2x+3} dx<br \\\/>&amp;= \\\\int \\\\color{red}{\\\\underbrace{\\\\color{black}{x}}_{\\\\large \\\\frac{u-3}{2}}} \\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\sqrt{2x+3}}}_{\\\\large u}}<br \\\/>\\\\color{red}{\\\\underbrace{\\\\color{black}{dx}}_{\\\\large \\\\frac{du}{2}}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{4}\\\\int (u-3)\\\\sqrt{u} du <br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the Substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{4} \\\\left(\\\\int{u\\\\sqrt{u}\\\\: du}\\\\: -\\\\: 3\\\\int \\\\sqrt{u} \\\\: du\\\\right)<br \\\/>&amp;\\\\color{gray}{\\\\text{[Distribute the Integrals]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{4} \\\\left(\\\\int{u^\\\\frac{3}{2}\\\\: du}\\\\: -\\\\: 3\\\\int u^\\\\frac{1}{2} \\\\: du\\\\right)<br \\\/>&amp;\\\\color{gray}{\\\\text{[Use power rules to simplify]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{4}\\\\left(\\\\frac{u^\\\\frac{5}{2}}{\\\\frac{5}{2}} &#8211; \\\\frac{u^\\\\frac{3}{2}}{\\\\frac{3}{2}}\\u00a0\\\\right) + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the Power Rule]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{4}\\\\left[\\\\left(\\\\frac{2}{5}\\\\right)u^\\\\frac{5}{2} &#8211; \\\\left(\\\\frac{2}{3}\\\\right)u^\\\\frac{3}{2}\\u00a0\\\\right] + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Simplify the fractions]}} \\\\\\\\[5pt]<br \\\/>&amp; = \\\\frac{1}{10}u^\\\\frac{5}{2}\\\\: &#8211; \\\\frac{1}{6}u^\\\\frac{3}{2} + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Distribute the outside fraction]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{1}{10}(2x+3)^\\\\frac{5}{2}\\\\: &#8211; \\\\frac{1}{6}(2x+3)^\\\\frac{3}{2} + C}<br \\\/>&amp;\\\\color{gray}{\\\\text{[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<div class=\"elementor-element elementor-element-0b07392 elementor-widget elementor-widget-toggle\" data-id=\"0b07392\" 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-1151\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1151\" 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-1151\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1151\"><p><span style=\"font-size: 18.6667px;\">Let \\(u=2x+1\\), \\(x = \\frac{u-1}{2}\\), and \\(du=2dx\\). Substituting these values into the integral leads to the following:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\frac{x}{2x+1} dx<br \/>&amp;= \\int \\left( \\frac{\\color{red}{\\frac{u-1}{2}}}{\\color{red}{u}}\\right) \\left( \\color{red}{\\frac{du}{2}}\\right)<br \/>&amp;\\color{gray}{\\text{[Apply the Substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{4} \\int\\frac{u-1}{u} du<br \/>&amp;\\color{gray}{\\text{[Rearrange the intergand]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{4} \\left( \\int \\frac{u}{u} du \\: -\\int \\frac{du}{u} \\right)<br \/>&amp;\\color{gray}{\\text{[Separate the integrals]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{4}\\left[ u &#8211; \\texttt{ln}|u| \\right] + C<br \/>&amp;\\color{gray}{\\text{[Simplify the first integrand and integrate]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{4} \\left[ 2x+1 &#8211; \\texttt{ln}|2x+1| \\right] + C<br \/>&amp;\\color{gray}{\\text{[Substitute back to x]}} \\\\[5pt]<br \/>&amp;= \\frac{x}{2} + \\frac{1}{4} -\\frac{1}{4} \\texttt{ln}|2x+1|+ C<br \/>&amp;\\color{gray}{\\text{[Distribute the fraction]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{x}{2}\u00a0 -\\frac{1}{4} \\texttt{ln}|2x+1|+ C}<br \/>&amp;\\color{gray}{\\text{[Include the fraction as part of the C constant]}} \\\\[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 3: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Let \\\\(u=2x+1\\\\), \\\\(x = \\\\frac{u-1}{2}\\\\), and \\\\(du=2dx\\\\). Substituting these values into the integral leads to the following:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\frac{x}{2x+1} dx<br \\\/>&amp;= \\\\int \\\\left( \\\\frac{\\\\color{red}{\\\\frac{u-1}{2}}}{\\\\color{red}{u}}\\\\right) \\\\left( \\\\color{red}{\\\\frac{du}{2}}\\\\right)<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the Substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{4} \\\\int\\\\frac{u-1}{u} du<br \\\/>&amp;\\\\color{gray}{\\\\text{[Rearrange the intergand]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{4} \\\\left( \\\\int \\\\frac{u}{u} du \\\\: -\\\\int \\\\frac{du}{u} \\\\right)<br \\\/>&amp;\\\\color{gray}{\\\\text{[Separate the integrals]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{4}\\\\left[ u &#8211; \\\\texttt{ln}|u| \\\\right] + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Simplify the first integrand and integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{4} \\\\left[ 2x+1 &#8211; \\\\texttt{ln}|2x+1| \\\\right] + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Substitute back to x]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{x}{2} + \\\\frac{1}{4} -\\\\frac{1}{4} \\\\texttt{ln}|2x+1|+ C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Distribute the fraction]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{x}{2}\\u00a0 -\\\\frac{1}{4} \\\\texttt{ln}|2x+1|+ C}<br \\\/>&amp;\\\\color{gray}{\\\\text{[Include the fraction as part of the C constant]}} \\\\\\\\[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-eacc7dd elementor-widget elementor-widget-toggle\" data-id=\"eacc7dd\" 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-2461\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2461\" 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-2461\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2461\"><p><span style=\"font-size: 18.6667px;\">Let \\(u=x^2\\), and \\(du=2xdx\\). Substituting these values into the integral leads to the following:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\frac{x^3}{x^2-4} dx<br \/>&amp;= \\int \\frac{x^2 \\cdot x}{x^2-4} dx<br \/>&amp;\\color{gray}{\\text{[Rewrite the integrand]}} \\\\[5pt]<br \/>&amp;= \\int \\left( \\frac{\\color{red}{u}}{\\color{red}{u-4}} \\right) \\left( \\color{red}{\\frac{du}{2}} \\right)<br \/>&amp;\\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\int \\frac{u}{u-4} du<br \/>&amp;\\color{gray}{\\text{[Rewrite the integrand]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\int \\frac{u-4+4}{u-4} du<br \/>&amp;\\color{gray}{\\text{[Add 0 by doing 4 &#8211; 4]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\left( \\int \\frac{u-4}{u-4} du\\: +4\\int \\frac{1}{u-4} du \\right)<br \/>&amp;\\color{gray}{\\text{[Separate the integrals strategically]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\left( \\int du\\: +4\\int \\frac{1}{u-4} du \\right)<br \/>&amp;\\color{gray}{\\text{[Simplify the first integrand]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} ( u\\: +4\\texttt{ln}|u-4|) + C<br \/>&amp;\\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} ( x^2\\: +4\\texttt{ln}|x^2-4|) + C<br \/>&amp;\\color{gray}{\\text{[Substitute back]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{x^2}{2}\\: +2\\texttt{ln}|x^2-4| + C}<br \/>&amp;\\color{gray}{\\text{[Distribute the fraction]}} \\\\[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\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Let \\\\(u=x^2\\\\), and \\\\(du=2xdx\\\\). Substituting these values into the integral leads to the following:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\frac{x^3}{x^2-4} dx<br \\\/>&amp;= \\\\int \\\\frac{x^2 \\\\cdot x}{x^2-4} dx<br \\\/>&amp;\\\\color{gray}{\\\\text{[Rewrite the integrand]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\int \\\\left( \\\\frac{\\\\color{red}{u}}{\\\\color{red}{u-4}} \\\\right) \\\\left( \\\\color{red}{\\\\frac{du}{2}} \\\\right)<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\int \\\\frac{u}{u-4} du<br \\\/>&amp;\\\\color{gray}{\\\\text{[Rewrite the integrand]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\int \\\\frac{u-4+4}{u-4} du<br \\\/>&amp;\\\\color{gray}{\\\\text{[Add 0 by doing 4 &#8211; 4]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\left( \\\\int \\\\frac{u-4}{u-4} du\\\\: +4\\\\int \\\\frac{1}{u-4} du \\\\right)<br \\\/>&amp;\\\\color{gray}{\\\\text{[Separate the integrals strategically]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\left( \\\\int du\\\\: +4\\\\int \\\\frac{1}{u-4} du \\\\right)<br \\\/>&amp;\\\\color{gray}{\\\\text{[Simplify the first integrand]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} ( u\\\\: +4\\\\texttt{ln}|u-4|) + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} ( x^2\\\\: +4\\\\texttt{ln}|x^2-4|) + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Substitute back]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{x^2}{2}\\\\: +2\\\\texttt{ln}|x^2-4| + C}<br \\\/>&amp;\\\\color{gray}{\\\\text{[Distribute the fraction]}} \\\\\\\\[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-15544be elementor-widget elementor-widget-toggle\" data-id=\"15544be\" 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-2231\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2231\" 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-2231\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2231\"><p><span style=\"font-size: 18.6667px;\">Let \\(u=2x^3+1\\), and \\(du=6x^2dx\\). Substituting these values into the integral leads to the following:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int x^2\\texttt{e}^{2x^3+1} dx<br \/>&amp;= \\int \\texttt{e}^{2x^3+1}(x^2dx)<br \/>&amp;\\color{gray}{\\text{[Rewrite the integrand]}} \\\\[5pt]<br \/>&amp;= \\int \\texttt{e}^\\color{red}{u} \\left( \\color{red}{\\frac{du}{6}} \\right)<br \/>&amp;\\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{6}\\int \\texttt{e}^udu<br \/>&amp;\\color{gray}{\\text{[Rewrite the integrand]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{6} \\texttt{e}^u\\:+ C<br \/>&amp;\\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{1}{6} \\texttt{e}^{2x^3+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 5: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Let \\\\(u=2x^3+1\\\\), and \\\\(du=6x^2dx\\\\). Substituting these values into the integral leads to the following:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int x^2\\\\texttt{e}^{2x^3+1} dx<br \\\/>&amp;= \\\\int \\\\texttt{e}^{2x^3+1}(x^2dx)<br \\\/>&amp;\\\\color{gray}{\\\\text{[Rewrite the integrand]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\int \\\\texttt{e}^\\\\color{red}{u} \\\\left( \\\\color{red}{\\\\frac{du}{6}} \\\\right)<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{6}\\\\int \\\\texttt{e}^udu<br \\\/>&amp;\\\\color{gray}{\\\\text{[Rewrite the integrand]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{6} \\\\texttt{e}^u\\\\:+ C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{1}{6} \\\\texttt{e}^{2x^3+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-a3a57eb elementor-widget elementor-widget-toggle\" data-id=\"a3a57eb\" 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-1711\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1711\" 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-1711\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1711\"><p><span style=\"font-size: 18.6667px;\">Let \\(u=\\texttt{e}^{2x}\\), and \\(du=2\\texttt{e}^{2x}dx\\). Substituting these values into the integral leads to the following:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\frac{\\texttt{e}^{2x}}{1+e^{2x}} dx<br \/>&amp;= \\int \\frac{\\color{red}{\\frac{du}{2}}}{\\color{red}{1+u}}<br \/>&amp;\\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\int \\frac{1}{1+u} du<br \/>&amp;\\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2}\\texttt{ln}|1+u|\\: + C &amp;\\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{1}{2}\\texttt{ln}|1+e^{2x}|\\: + 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 6: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Let \\\\(u=\\\\texttt{e}^{2x}\\\\), and \\\\(du=2\\\\texttt{e}^{2x}dx\\\\). Substituting these values into the integral leads to the following:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\frac{\\\\texttt{e}^{2x}}{1+e^{2x}} dx<br \\\/>&amp;= \\\\int \\\\frac{\\\\color{red}{\\\\frac{du}{2}}}{\\\\color{red}{1+u}}<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\int \\\\frac{1}{1+u} du<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2}\\\\texttt{ln}|1+u|\\\\: + C &amp;\\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{1}{2}\\\\texttt{ln}|1+e^{2x}|\\\\: + 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-3f0b091 elementor-widget elementor-widget-toggle\" data-id=\"3f0b091\" 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-6611\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-6611\" 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-6611\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-6611\"><p><span style=\"font-size: 18.6667px;\">Let \\(u=\\texttt{ln}(x)\\), and \\(du=\\frac{dx}{x}\\). Substituting these values into the integral leads to the following:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int^\\texttt{e}_1 \\frac{\\texttt{ln}(x)}{x} dx<br \/>&amp;= \\int \\frac{\\color{red}{\\frac{du}{2}}}{\\color{red}{1+u}}<br \/>&amp;\\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\int \\frac{1}{1+u} du<br \/>&amp;\\color{gray}{\\text{[Rewrite the integrand]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2}\\texttt{ln}|1+u|\\: + C<br \/>&amp;\\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{1}{2}\\texttt{ln}|1+\\texttt{e}^{2x}|\\: + 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 7: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Let \\\\(u=\\\\texttt{ln}(x)\\\\), and \\\\(du=\\\\frac{dx}{x}\\\\). Substituting these values into the integral leads to the following:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int^\\\\texttt{e}_1 \\\\frac{\\\\texttt{ln}(x)}{x} dx<br \\\/>&amp;= \\\\int \\\\frac{\\\\color{red}{\\\\frac{du}{2}}}{\\\\color{red}{1+u}}<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\int \\\\frac{1}{1+u} du<br \\\/>&amp;\\\\color{gray}{\\\\text{[Rewrite the integrand]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2}\\\\texttt{ln}|1+u|\\\\: + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{1}{2}\\\\texttt{ln}|1+\\\\texttt{e}^{2x}|\\\\: + 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-b263745 elementor-widget elementor-widget-toggle\" data-id=\"b263745\" 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-1871\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1871\" 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-1871\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1871\"><p><span style=\"font-size: 18.6667px;\">Let \\(u=\\texttt{cos}(x)+1\\), and\u00a0<\/span><span style=\"font-size: 18.6667px;\">\\(du=-\\texttt{sin}(x)dx\\)<\/span><span style=\"font-size: 18.6667px;\">. Substituting these values into the integral leads to the following:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\frac{\\texttt{sin}(x)}{1+\\texttt{cos}(x)} dx<br \/>&amp;= \\int \\frac{\\color{red}{-1}}{\\color{red}{u}} du<br \/>&amp;\\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= -\\texttt{ln}|u|\\: + C<br \/>&amp;\\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\color{green}{-\\texttt{ln}|\\texttt{cos}(x)+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 8: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Let \\\\(u=\\\\texttt{cos}(x)+1\\\\), and\\u00a0<\\\/span><span style=\\\"font-size: 18.6667px;\\\">\\\\(du=-\\\\texttt{sin}(x)dx\\\\)<\\\/span><span style=\\\"font-size: 18.6667px;\\\">. Substituting these values into the integral leads to the following:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\frac{\\\\texttt{sin}(x)}{1+\\\\texttt{cos}(x)} dx<br \\\/>&amp;= \\\\int \\\\frac{\\\\color{red}{-1}}{\\\\color{red}{u}} du<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= -\\\\texttt{ln}|u|\\\\: + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{-\\\\texttt{ln}|\\\\texttt{cos}(x)+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-a0ba4a5 elementor-widget elementor-widget-toggle\" data-id=\"a0ba4a5\" 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-1681\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1681\" 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-1681\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1681\"><p><span style=\"font-size: 18.6667px;\">Let \\(u=\\texttt{e}^x\\), and\u00a0<\/span><span style=\"font-size: 18.6667px;\">\\(du=\\texttt{e}^xdx\\)<\/span><span style=\"font-size: 18.6667px;\">. Substituting these values into the integral leads to the following:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\frac{\\texttt{e}^x}{1+\\texttt{e}^{2x}} dx<br \/>&amp;= \\int \\frac{\\color{red}{du}}{\\color{red}{1+u^2}} du<br \/>&amp;\\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\texttt{arctan}(u)\\: + C<br \/>&amp;\\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\texttt{arctan}(\\texttt{e}^x)\\:+ 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 9: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Let \\\\(u=\\\\texttt{e}^x\\\\), and\\u00a0<\\\/span><span style=\\\"font-size: 18.6667px;\\\">\\\\(du=\\\\texttt{e}^xdx\\\\)<\\\/span><span style=\\\"font-size: 18.6667px;\\\">. Substituting these values into the integral leads to the following:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\frac{\\\\texttt{e}^x}{1+\\\\texttt{e}^{2x}} dx<br \\\/>&amp;= \\\\int \\\\frac{\\\\color{red}{du}}{\\\\color{red}{1+u^2}} du<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\texttt{arctan}(u)\\\\: + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\texttt{arctan}(\\\\texttt{e}^x)\\\\:+ 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-568d528 elementor-widget elementor-widget-toggle\" data-id=\"568d528\" 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-9071\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-9071\" 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-9071\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-9071\"><p><span style=\"font-size: 18.6667px;\">Transform the current integrand using algebraic manipulations:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\frac{1}{\\texttt{e}^x+\\texttt{e}^{-x}} dx<br \/>&amp;= \\int \\frac{1}{1+\\frac{1}{\\texttt{e}^x}} dx<br \/>&amp;\\color{gray}{\\text{[Use power rules]}} \\\\[5pt]<br \/>&amp;= \\int \\frac{1}{\\frac{\\texttt{e}^{2x}+1}{\\texttt{e}^x}} dx<br \/>&amp;\\color{gray}{\\text{[Perform the sum]}} \\\\[5pt]<br \/>&amp;= \\int \\frac{\\texttt{e}^x}{\\texttt{e}^{2x}+1} dx<br \/>&amp;\\color{gray}{\\text{[Reverse the fraction]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<p><span style=\"font-size: 18.6667px;\">\u00a0<\/span><\/p><p><span style=\"font-size: 18.6667px;\">Let \\(u=\\texttt{e}^x\\), and\u00a0<\/span><span style=\"font-size: 18.6667px;\">\\(du=\\texttt{e}^xdx\\)<\/span><span style=\"font-size: 18.6667px;\">. Substituting these values into the integral leads to the following:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\frac{\\texttt{e}^x}{\\texttt{e}^{2x}+1} dx<br \/>&amp;= \\int \\frac{\\color{red}{du}}{\\color{red}{u^2+1}}<br \/>&amp;\\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\texttt{arctan}(u)\\:+ C<br \/>&amp;\\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\texttt{arctan}(\\texttt{e}^x)\\: + 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 10: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Transform the current integrand using algebraic manipulations:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\frac{1}{\\\\texttt{e}^x+\\\\texttt{e}^{-x}} dx<br \\\/>&amp;= \\\\int \\\\frac{1}{1+\\\\frac{1}{\\\\texttt{e}^x}} dx<br \\\/>&amp;\\\\color{gray}{\\\\text{[Use power rules]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\int \\\\frac{1}{\\\\frac{\\\\texttt{e}^{2x}+1}{\\\\texttt{e}^x}} dx<br \\\/>&amp;\\\\color{gray}{\\\\text{[Perform the sum]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\int \\\\frac{\\\\texttt{e}^x}{\\\\texttt{e}^{2x}+1} dx<br \\\/>&amp;\\\\color{gray}{\\\\text{[Reverse the fraction]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p><span style=\\\"font-size: 18.6667px;\\\">\\u00a0<\\\/span><\\\/p><p><span style=\\\"font-size: 18.6667px;\\\">Let \\\\(u=\\\\texttt{e}^x\\\\), and\\u00a0<\\\/span><span style=\\\"font-size: 18.6667px;\\\">\\\\(du=\\\\texttt{e}^xdx\\\\)<\\\/span><span style=\\\"font-size: 18.6667px;\\\">. Substituting these values into the integral leads to the following:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\frac{\\\\texttt{e}^x}{\\\\texttt{e}^{2x}+1} dx<br \\\/>&amp;= \\\\int \\\\frac{\\\\color{red}{du}}{\\\\color{red}{u^2+1}}<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\texttt{arctan}(u)\\\\:+ C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\texttt{arctan}(\\\\texttt{e}^x)\\\\: + 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-b056ea8 elementor-widget elementor-widget-toggle\" data-id=\"b056ea8\" 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-1841\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1841\" 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-1841\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1841\"><p><span style=\"font-size: 18.6667px;\">Let \\(u=\\sqrt{x}\\), \\(du=\\frac{1}{2\\sqrt{x}}dx\\), and \\(x+1=u^2+1\\).<\/span><span style=\"font-size: 18.6667px;\">\u00a0<\/span><span style=\"font-size: 18.6667px;\">Substituting these values into the integral leads to the following:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\frac{1}{\\sqrt{x}(x+1)} dx<br \/>&amp;= \\int \\frac{\\color{red}{2 \\cdot du}}{\\color{red}{u^2+1}}<br \/>&amp;\\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= 2\\int \\frac{1}{u^2+1} du<br \/>&amp;\\color{gray}{\\text{[Rearrange the integrand]}} \\\\[5pt]<br \/>&amp;= 2\\texttt{arctan}(u)\\: + C<br \/>&amp;\\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\color{green}{2\\texttt{arctan}(\\sqrt{x})\\: + 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\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Let \\\\(u=\\\\sqrt{x}\\\\), \\\\(du=\\\\frac{1}{2\\\\sqrt{x}}dx\\\\), and \\\\(x+1=u^2+1\\\\).<\\\/span><span style=\\\"font-size: 18.6667px;\\\">\\u00a0<\\\/span><span style=\\\"font-size: 18.6667px;\\\">Substituting these values into the integral leads to the following:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\frac{1}{\\\\sqrt{x}(x+1)} dx<br \\\/>&amp;= \\\\int \\\\frac{\\\\color{red}{2 \\\\cdot du}}{\\\\color{red}{u^2+1}}<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= 2\\\\int \\\\frac{1}{u^2+1} du<br \\\/>&amp;\\\\color{gray}{\\\\text{[Rearrange the integrand]}} \\\\\\\\[5pt]<br \\\/>&amp;= 2\\\\texttt{arctan}(u)\\\\: + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{2\\\\texttt{arctan}(\\\\sqrt{x})\\\\: + 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-125d2be elementor-widget elementor-widget-toggle\" data-id=\"125d2be\" 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-1921\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1921\" 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-1921\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1921\"><p><span style=\"font-size: 18.6667px;\">Let \\(u=x^2\\), and \\(du=2xdx\\).<\/span><span style=\"font-size: 18.6667px;\">\u00a0<\/span><span style=\"font-size: 18.6667px;\">Substituting these values into the integral leads to the following:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\frac{x}{\\sqrt{1-x^4}} dx<br \/>&amp;= \\int \\frac{\\color{red}{\\frac{du}{2}}}{\\sqrt{1-\\color{red}{u^2}}}<br \/>&amp;\\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\int \\frac{1}{\\sqrt{1-u^2}} du<br \/>&amp;\\color{gray}{\\text{[Rearrange the integrand]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\texttt{arcsin}(u)\\: + C<br \/>&amp;\\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{1}{2}\\texttt{arcsin}(x^2)\\: + 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 12: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Let \\\\(u=x^2\\\\), and \\\\(du=2xdx\\\\).<\\\/span><span style=\\\"font-size: 18.6667px;\\\">\\u00a0<\\\/span><span style=\\\"font-size: 18.6667px;\\\">Substituting these values into the integral leads to the following:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\frac{x}{\\\\sqrt{1-x^4}} dx<br \\\/>&amp;= \\\\int \\\\frac{\\\\color{red}{\\\\frac{du}{2}}}{\\\\sqrt{1-\\\\color{red}{u^2}}}<br \\\/>&amp;\\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\int \\\\frac{1}{\\\\sqrt{1-u^2}} du<br \\\/>&amp;\\\\color{gray}{\\\\text{[Rearrange the integrand]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\texttt{arcsin}(u)\\\\: + C<br \\\/>&amp;\\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{1}{2}\\\\texttt{arcsin}(x^2)\\\\: + 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-424305c elementor-widget elementor-widget-toggle\" data-id=\"424305c\" 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-6941\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-6941\" 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-6941\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-6941\"><p><span style=\"font-size: 18.6667px;\">Transform the integrand using algebraic manipulations:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\frac{x^2}{x^2+1} dx<br \/>&amp;= \\int \\frac{x^2+1-1}{x^2+1} dx<br \/>&amp;\\color{gray}{\\text{[Add 0 by doing 1-1]}} \\\\[5pt]<br \/>&amp;= \\int \\frac{x^2+1}{x^2+1} dx\\:- \\int \\frac{1}{x^2+1} dx<br \/>&amp;\\color{gray}{\\text{[Separate the integrals]}} \\\\[5pt]<br \/>&amp;= \\int dx\\:- \\int \\frac{1}{x^2+1} dx<br \/>&amp;\\color{gray}{\\text{[Simplify the first integrand]}} \\\\[5pt]<br \/>&amp;= \\color{green}{x\\:- \\texttt{arctan}(x)\\:+ C}<br \/>&amp;\\color{gray}{\\text{[Integrate]}} \\\\[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 13: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Transform the integrand using algebraic manipulations:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\frac{x^2}{x^2+1} dx<br \\\/>&amp;= \\\\int \\\\frac{x^2+1-1}{x^2+1} dx<br \\\/>&amp;\\\\color{gray}{\\\\text{[Add 0 by doing 1-1]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\int \\\\frac{x^2+1}{x^2+1} dx\\\\:- \\\\int \\\\frac{1}{x^2+1} dx<br \\\/>&amp;\\\\color{gray}{\\\\text{[Separate the integrals]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\int dx\\\\:- \\\\int \\\\frac{1}{x^2+1} dx<br \\\/>&amp;\\\\color{gray}{\\\\text{[Simplify the first integrand]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{x\\\\:- \\\\texttt{arctan}(x)\\\\:+ C}<br \\\/>&amp;\\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[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-f154c14 elementor-widget elementor-widget-button\" data-id=\"f154c14\" data-element_type=\"widget\" id=\"top\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"#\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Back To Top<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\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-fdb8651 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fdb8651\" 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-7adec0e\" data-id=\"7adec0e\" 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-9294114 elementor-widget elementor-widget-spacer\" data-id=\"9294114\" 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 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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 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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 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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 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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 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The number of partitions must be even. Virtual Lessons Need some additional <a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-simpsons\/\" class=\"read-more\">Read More &#8230;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":19541,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"elementor_header_footer","meta":{"footnotes":""},"class_list":["post-20092","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.6 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Integration Practice Problems - [Simpsons] | Online Math Tutoring LLC | Expert Math Tutor Near Me<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-simpsons\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration Practice Problems - [Simpsons] | Online Math Tutoring LLC | Expert Math Tutor Near Me\" \/>\n<meta property=\"og:description\" content=\"Practice Problems: Simpson&#8217;s Rule (1\/3) Also known as Simpson&#8217;s Rule is a numerical integration technique that improves upon the Trapezoidal Rule by utilizing the geometry of parabolic arcs. 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