{"id":19873,"date":"2022-02-23T01:32:02","date_gmt":"2022-02-23T01:32:02","guid":{"rendered":"https:\/\/onlinemathtutor.co\/?page_id=19873"},"modified":"2022-06-06T03:13:12","modified_gmt":"2022-06-06T03:13:12","slug":"integration-practice-problems-trigonometric-integrals","status":"publish","type":"page","link":"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-integrals\/","title":{"rendered":"Integration Practice Problems &#8211; [Trigonometric Integrals]"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"19873\" class=\"elementor elementor-19873\" 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: Trigonometric Integrals<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9ac37bb elementor-nav-menu__align-center elementor-nav-menu--stretch elementor-nav-menu__text-align-center animated-fast elementor-nav-menu--dropdown-tablet elementor-nav-menu--toggle elementor-nav-menu--burger elementor-widget elementor-widget-nav-menu\" data-id=\"9ac37bb\" data-element_type=\"widget\" data-settings=\"{&quot;submenu_icon&quot;:{&quot;value&quot;:&quot;&lt;i class=\\&quot;fas fa-pencil-alt\\&quot;&gt;&lt;\\\/i&gt;&quot;,&quot;library&quot;:&quot;fa-solid&quot;},&quot;full_width&quot;:&quot;stretch&quot;,&quot;_animation&quot;:&quot;none&quot;,&quot;layout&quot;:&quot;horizontal&quot;,&quot;toggle&quot;:&quot;burger&quot;}\" data-widget_type=\"nav-menu.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<nav aria-label=\"Menu\" class=\"elementor-nav-menu--main elementor-nav-menu__container elementor-nav-menu--layout-horizontal e--pointer-double-line e--animation-fast\">\n\t\t\t\t<ul id=\"menu-1-9ac37bb\" class=\"elementor-nav-menu\"><li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-19677\"><a href=\"https:\/\/onlinemathtutor.co\/index.php\/problem-set-integration\/\" class=\"elementor-item\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2021\/06\/Home-Icon-Blue.png\"  width=\"20\"><\/a><\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19577\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Integration Techniques<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19777\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">U-Substitution<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19576\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-usub\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19776\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-usub\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-19694\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-ibp\/\" class=\"elementor-sub-item\">Integration By Parts<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19898\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-ibp\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19956\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-ibp\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19885\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Trigonometric Integrals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19925\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-integrals\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19954\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-integrals\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19886\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Trigonometric Substitution<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19926\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-substitution\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19953\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-substitution\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19884\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Partial Fractions<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19924\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-partial-fractions\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19955\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-partialfractions\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19893\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Improper Integrals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19927\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-improper-integrals\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19966\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-improper-integrals\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19578\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Integration Applications<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19888\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Area<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19928\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-area\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20136\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-area\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19889\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Volumes [Nonrotational]<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19929\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-volume\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20582\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-substitution\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19890\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Volume [Rotational]<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19931\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-rotational-volumes\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20160\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-rotational-volume\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19891\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Surface Area<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19930\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-surface-area\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20165\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-surface-area\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19892\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Arc Length<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20177\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-arc-length\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20176\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-arc-length\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19579\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Numerical Integration<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19894\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Reimann Sums<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20110\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-riemann-sums\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20128\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-riemann-sum\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19895\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Trapezoidal Rule<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20109\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trapezoidal\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20138\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trapezoidal\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19896\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Simpson&#8217;s Rule<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20108\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-simpsons\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20137\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-simpsons\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t<div class=\"elementor-menu-toggle\" role=\"button\" tabindex=\"0\" aria-label=\"Menu Toggle\" aria-expanded=\"false\">\n\t\t\t<i aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--open eicon-menu-bar\"><\/i><i aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--close eicon-close\"><\/i>\t\t<\/div>\n\t\t\t\t\t<nav class=\"elementor-nav-menu--dropdown elementor-nav-menu__container\" aria-hidden=\"true\">\n\t\t\t\t<ul id=\"menu-2-9ac37bb\" class=\"elementor-nav-menu\"><li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-19677\"><a href=\"https:\/\/onlinemathtutor.co\/index.php\/problem-set-integration\/\" class=\"elementor-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2021\/06\/Home-Icon-Blue.png\"  width=\"20\"><\/a><\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19577\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Integration Techniques<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19777\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">U-Substitution<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19576\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-usub\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19776\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-usub\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-19694\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-ibp\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Integration By Parts<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19898\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-ibp\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19956\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-ibp\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19885\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Trigonometric Integrals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19925\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-integrals\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19954\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-integrals\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19886\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Trigonometric Substitution<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19926\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-substitution\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19953\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-substitution\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19884\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Partial Fractions<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19924\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-partial-fractions\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19955\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-partialfractions\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19893\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Improper Integrals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19927\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-improper-integrals\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19966\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-improper-integrals\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19578\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Integration Applications<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19888\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Area<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19928\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-area\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20136\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-area\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19889\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Volumes [Nonrotational]<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19929\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-volume\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20582\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-substitution\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19890\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Volume [Rotational]<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19931\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-rotational-volumes\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20160\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-rotational-volume\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19891\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Surface Area<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19930\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-surface-area\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20165\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-surface-area\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19892\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Arc Length<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20177\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-arc-length\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20176\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-arc-length\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19579\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Numerical Integration<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19894\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Reimann Sums<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20110\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-riemann-sums\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20128\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-riemann-sum\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19895\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Trapezoidal Rule<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20109\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trapezoidal\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20138\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trapezoidal\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19896\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Simpson&#8217;s Rule<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20108\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-simpsons\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20137\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-simpsons\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a75a15d elementor-section-stretched elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a75a15d\" data-element_type=\"section\" data-settings=\"{&quot;stretch_section&quot;:&quot;section-stretched&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-555d8f5\" data-id=\"555d8f5\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-356b90a elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"356b90a\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6f95eaf elementor-widget elementor-widget-text-editor\" data-id=\"6f95eaf\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"text-align: justify;\"><span style=\"font-size: 20px;\">When integrating <em>products<\/em> of trigonometric functions, the general practice involves applying the trigonometric versions of the Pythagorean Theorem such as\u00a0<\/span><\/p><p style=\"text-align: center;\"><span style=\"font-size: 20px;\">\\(\\texttt{sin}^2(\\theta)+\\texttt{cos}^2(\\theta)=1\\)\u00a0 \u00a0 or\u00a0 \u00a0 \\(\\texttt{tan}^2(\\theta)+1=\\texttt{sec}^2(\\theta)\\)<\/span><\/p><p style=\"text-align: justify;\"><span style=\"font-size: 20px;\">in conjunction with an appropriate u-substitution. If the powers both even then you may need to consider applying a Power Reduction Identitiy (Half-Angle Idenitty).\u00a0<\/span><\/p><p style=\"text-align: center;\"><span style=\"font-size: 20px;\">\\(\\texttt{sin}(2\\theta)\\frac{1-\\texttt{cos}(2\\theta)}{2}\\)\u00a0 \u00a0 or\u00a0 \\(\\texttt{cos}(2\\theta)\\frac{1+\\texttt{cos}(2\\theta)}{2}\\)<\/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-trigonometric-integrals\/?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-trigonometric-integrals\/?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;\">Answer the following questions by employing the Pythagorean Theorem or the Power Reduction identities.\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-352efd9f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"352efd9f\" 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-513f6b43\" data-id=\"513f6b43\" 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-3295de5c elementor-widget elementor-widget-text-editor\" data-id=\"3295de5c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ol start=\"1\"><li>\\(\\displaystyle \\int\u00a0\\texttt{sin}^2(x)\\texttt{cos}^4(x) dx \\)<br \/><br \/><\/li><li>\\(\\displaystyle \\int \\texttt{sin}^3(x)\\texttt{cos}^4(x)\u00a0dx \\)<br \/><br \/><\/li><li>\\(\\displaystyle \\int \\texttt{sin}^3(x)\u00a0dx \\)<br \/><br \/><\/li><li>\\(\\displaystyle \\int^{\\pi}_{-\\pi} \\texttt{sin}(3x)\\texttt{cos}(2x)\u00a0dx \\)<\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-2e5c3e39\" data-id=\"2e5c3e39\" 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-1f9bd4ba elementor-widget elementor-widget-text-editor\" data-id=\"1f9bd4ba\" 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=\"5\"><li>\\(\\displaystyle \\int^{\\frac{\\pi}{4}}_{0}\u00a0 \\texttt{sin}^3(x) dx \\) <br \/><br \/><\/li><li>\\(\\displaystyle \\int\u00a0\\texttt{sec}(4x) dx \\)<br \/><br \/><\/li><li>\\(\\displaystyle \\int \\texttt{tan}^3(3x)\\texttt{sec}^2(3x)\u00a0dx \\)<br \/><br \/><\/li><li>\\(\\displaystyle \\int \\texttt{tan}^4(x)\\texttt{sec}^2(x)\u00a0dx \\)<\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-1cb1678c\" data-id=\"1cb1678c\" 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-fc17ee1 elementor-widget elementor-widget-text-editor\" data-id=\"fc17ee1\" 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=\"9\"><li>\\(\\displaystyle \\int\u00a0\\frac{1+\\texttt{cos}(2x)}{1-\\texttt{cos}(2x)} dx \\)<br \/><br \/><\/li><li>\\(\\displaystyle \\int\u00a0\\sqrt{1-\\texttt{cos}(\\theta)} d\\theta \\)<\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-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-362a8369 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"362a8369\" 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-3f49ca2a\" data-id=\"3f49ca2a\" 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-34917060 elementor-widget elementor-widget-toggle\" data-id=\"34917060\" 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-8811\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-8811\" 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-8811\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-8811\"><p><span style=\"font-size: 18.6667px;\">Use the double angle identity for sines: \\(2\\texttt{sin}(x)\\texttt{cos}(x) = \\texttt{sin}(2x)\\), so \\(\\texttt{sin}(x)\\texttt{cos}(x)=\\frac{1}{2}\\texttt{sin}(2x)\\):<\/span><\/p><p>\u00a0<\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\texttt{sin}^2(x)\\texttt{cos}^4(x) dx<br \/>&amp;= \\int [\\color{red}{\\underbrace{\\color{black}{\\texttt{sin}(x)\\texttt{cos}(x)}}_{\\large \\frac{1}{2}\\texttt{sin}(2x)}}]^2 \\texttt{cos}^2(x)dx <br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{4}\\int \\texttt{sin}^2(2x)\\texttt{cos}^2(x)dx<br \/>&amp; \\color{gray}{\\text{[Rearrange the expression]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><p>\u00a0<\/p><p><span style=\"font-size: 18.6667px;\">Apply the power reduction identity for cosines: \\(\\texttt{cos}^2(x) = \\frac{1\\:+ \\texttt{cos}(2x)}{2}\\). Substituting yields:<\/span><\/p><p>\u00a0<\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\frac{1}{4}\\int \\texttt{sin}^2(2x)\\texttt{cos}^2(x)dx<br \/>&amp;= \\frac{1}{4}\\int [\\texttt{sin}^2(2x)] \\left[ \\color{red}{\\underbrace{\\color{black}{\\frac{1\\:+ \\texttt{cos}(2x)}{2}}}_{\\texttt{cos}^2(x)}}dx \\right]<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{8}\\left( \\int \\texttt{sin}^2(2x)dx\\:+ \\int \\texttt{sin}^2(2x)\\texttt{cos}(2x)dx \\right)<br \/>&amp; \\color{gray}{\\text{[Rearrange the expression]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><p>\u00a0<\/p><p><span style=\"font-size: 18.6667px;\">Use a quick U-sub. Let \\(u=2x\\), so \\(du=2dx\\). Substituting yields:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\frac{1}{4}\\int \\texttt{sin}^2(2x)\\texttt{cos}^2(x)dx<br \/>&amp;= \\frac{1}{8}\\left( \\int \\texttt{sin}^2(\\color{red}{\\underbrace{\\color{black}{2x}}_{\\large u}}) \\color{red}{\\underbrace{\\color{black}{dx}}_{\\large \\frac{du}{2}}} \\:+ \\int \\texttt{sin}^2(\\color{red}{\\underbrace{\\color{black}{2x}}_{\\large u}})\\texttt{cos}(\\color{red}{\\underbrace{\\color{black}{2x}}_{\\large u}}) \\color{red}{\\underbrace{\\color{black}{dx}}_{\\large \\frac{du}{2}}} \\right)<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{16}\\left( \\int \\texttt{sin}^2(u)du\\:+ \\int \\texttt{sin}^2(u)\\texttt{cos}(u)du \\right)<br \/>&amp; \\color{gray}{\\text{[Factor and rearrange the expression]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><p>\u00a0<\/p><p><span style=\"font-size: 18.6667px;\">Apply the power reduction identity for sines in the first integrand: \\(\\texttt{sin}^2(x) = \\frac{1\\:- \\texttt{cos}(2x)}{2}\\). Substituting yields:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\frac{1}{16}\\left( \\int \\texttt{sin}^2(u)du\\:+ \\int \\texttt{sin}^2(u)\\texttt{cos}(u)du \\right)<br \/>&amp;= \\frac{1}{16}\\left( \\int \\color{red}{\\underbrace{\\color{black}{\\texttt{sin}^2(u)}}_{\\large \\frac{1-\\texttt{cos}(2u)}{2}}} \\:+ \\int \\texttt{sin}^2(u)\\texttt{cos}(u)du \\right)<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{16}\\left( \\frac{1}{2}\\int du\\:- \\frac{1}{2}\\int \\texttt{cos}(2u)du \\:+ \\int \\texttt{sin}^2(u)\\texttt{cos}(u)du \\right)<br \/>&amp; \\color{gray}{\\text{[Distribute the first integral]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{32}u\\:- \\frac{1}{64}\\texttt{sin}(2u)\\:+ \\frac{1}{16}\\int \\texttt{sin}^2(u)\\texttt{cos}(u)du<br \/>&amp; \\color{gray}{\\text{[Integrate (direct sub and u-sub)]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{32}(2x)\\:- \\frac{1}{64}\\texttt{sin}(2 \\cdot 2x)\\:+ \\frac{1}{16}\\int \\texttt{sin}^2(u)\\texttt{cos}(u)du<br \/>&amp; \\color{gray}{\\text{[Substitute back to x]}} \\\\[5pt]<br \/>&amp;= \\frac{x}{16}\\:- \\frac{1}{64}\\texttt{sin}(4x)\\:+ \\frac{1}{16} \\color{red}{\\int \\texttt{sin}^2(u)\\texttt{cos}(u)du}<br \/>&amp; \\color{gray}{\\text{[Simplify]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><p>\u00a0<\/p><p><span style=\"font-size: 18.6667px;\">Use a U-sub for the third integral. Let \\(v=\\texttt{sin}(u)\\), so \\(dv=\\texttt{cos}(u)du\\):<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\color{red}{\\int \\texttt{sin}^2(u)\\texttt{cos}(u)du}<br \/>&amp;= \\int \\color{red}{\\underbrace{\\color{black}{\\texttt{sin}^2(u)}}_{\\large v^2}} \\color{red}{\\underbrace{\\color{black}{\\texttt{cos}(u)du}}_{\\large dv}}\u00a0<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\int v^2dv = \\frac{1}{3} v^3\\:+ C<br \/>&amp; \\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{3} \\texttt{sin}^3(u)\\:+ C<br \/>&amp; \\color{gray}{\\text{[Substitute back to u]}} \\\\[5pt]<br \/>&amp;= \\color{red}{\\frac{1}{3} \\texttt{sin}^3(2x)\\:+ C}<br \/>&amp; \\color{gray}{\\text{[Substitute back to x]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><p>\u00a0<\/p><p><span style=\"font-size: 18.6667px;\">Combine previous results into one expression:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\texttt{sin}^2(x)\\texttt{cos}^4(x) dx<br \/>&amp;= \\frac{x}{16}\\:- \\frac{1}{64}\\texttt{sin}(4x)\\:+ \\frac{1}{16} \\color{red}{\\int \\texttt{sin}^2(u)\\texttt{cos}(u)du}<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{x}{16}\\:- \\frac{1}{64}\\texttt{sin}(4x)\\:+ \\frac{1}{16} \\left(\\color{red}{\\frac{1}{3} \\texttt{sin}^3(2x)}\\right)\\:+ C<br \/>&amp; \\color{gray}{\\text{[Rearrange the expression]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{1}{48}\\texttt{sin}^3(2x)\\:- \\frac{1}{64}\\texttt{sin}(4x)\\:+ \\frac{x}{16}\\:+ C}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 1: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Use the double angle identity for sines: \\\\(2\\\\texttt{sin}(x)\\\\texttt{cos}(x) = \\\\texttt{sin}(2x)\\\\), so \\\\(\\\\texttt{sin}(x)\\\\texttt{cos}(x)=\\\\frac{1}{2}\\\\texttt{sin}(2x)\\\\):<\\\/span><\\\/p><p>\\u00a0<\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\texttt{sin}^2(x)\\\\texttt{cos}^4(x) dx<br \\\/>&amp;= \\\\int [\\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{sin}(x)\\\\texttt{cos}(x)}}_{\\\\large \\\\frac{1}{2}\\\\texttt{sin}(2x)}}]^2 \\\\texttt{cos}^2(x)dx <br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{4}\\\\int \\\\texttt{sin}^2(2x)\\\\texttt{cos}^2(x)dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Rearrange the expression]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><p>\\u00a0<\\\/p><p><span style=\\\"font-size: 18.6667px;\\\">Apply the power reduction identity for cosines: \\\\(\\\\texttt{cos}^2(x) = \\\\frac{1\\\\:+ \\\\texttt{cos}(2x)}{2}\\\\). Substituting yields:<\\\/span><\\\/p><p>\\u00a0<\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\frac{1}{4}\\\\int \\\\texttt{sin}^2(2x)\\\\texttt{cos}^2(x)dx<br \\\/>&amp;= \\\\frac{1}{4}\\\\int [\\\\texttt{sin}^2(2x)] \\\\left[ \\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\frac{1\\\\:+ \\\\texttt{cos}(2x)}{2}}}_{\\\\texttt{cos}^2(x)}}dx \\\\right]<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{8}\\\\left( \\\\int \\\\texttt{sin}^2(2x)dx\\\\:+ \\\\int \\\\texttt{sin}^2(2x)\\\\texttt{cos}(2x)dx \\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Rearrange the expression]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><p>\\u00a0<\\\/p><p><span style=\\\"font-size: 18.6667px;\\\">Use a quick U-sub. Let \\\\(u=2x\\\\), so \\\\(du=2dx\\\\). Substituting yields:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\frac{1}{4}\\\\int \\\\texttt{sin}^2(2x)\\\\texttt{cos}^2(x)dx<br \\\/>&amp;= \\\\frac{1}{8}\\\\left( \\\\int \\\\texttt{sin}^2(\\\\color{red}{\\\\underbrace{\\\\color{black}{2x}}_{\\\\large u}}) \\\\color{red}{\\\\underbrace{\\\\color{black}{dx}}_{\\\\large \\\\frac{du}{2}}} \\\\:+ \\\\int \\\\texttt{sin}^2(\\\\color{red}{\\\\underbrace{\\\\color{black}{2x}}_{\\\\large u}})\\\\texttt{cos}(\\\\color{red}{\\\\underbrace{\\\\color{black}{2x}}_{\\\\large u}}) \\\\color{red}{\\\\underbrace{\\\\color{black}{dx}}_{\\\\large \\\\frac{du}{2}}} \\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{16}\\\\left( \\\\int \\\\texttt{sin}^2(u)du\\\\:+ \\\\int \\\\texttt{sin}^2(u)\\\\texttt{cos}(u)du \\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Factor and rearrange the expression]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><p>\\u00a0<\\\/p><p><span style=\\\"font-size: 18.6667px;\\\">Apply the power reduction identity for sines in the first integrand: \\\\(\\\\texttt{sin}^2(x) = \\\\frac{1\\\\:- \\\\texttt{cos}(2x)}{2}\\\\). Substituting yields:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\frac{1}{16}\\\\left( \\\\int \\\\texttt{sin}^2(u)du\\\\:+ \\\\int \\\\texttt{sin}^2(u)\\\\texttt{cos}(u)du \\\\right)<br \\\/>&amp;= \\\\frac{1}{16}\\\\left( \\\\int \\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{sin}^2(u)}}_{\\\\large \\\\frac{1-\\\\texttt{cos}(2u)}{2}}} \\\\:+ \\\\int \\\\texttt{sin}^2(u)\\\\texttt{cos}(u)du \\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{16}\\\\left( \\\\frac{1}{2}\\\\int du\\\\:- \\\\frac{1}{2}\\\\int \\\\texttt{cos}(2u)du \\\\:+ \\\\int \\\\texttt{sin}^2(u)\\\\texttt{cos}(u)du \\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Distribute the first integral]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{32}u\\\\:- \\\\frac{1}{64}\\\\texttt{sin}(2u)\\\\:+ \\\\frac{1}{16}\\\\int \\\\texttt{sin}^2(u)\\\\texttt{cos}(u)du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate (direct sub and u-sub)]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{32}(2x)\\\\:- \\\\frac{1}{64}\\\\texttt{sin}(2 \\\\cdot 2x)\\\\:+ \\\\frac{1}{16}\\\\int \\\\texttt{sin}^2(u)\\\\texttt{cos}(u)du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back to x]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{x}{16}\\\\:- \\\\frac{1}{64}\\\\texttt{sin}(4x)\\\\:+ \\\\frac{1}{16} \\\\color{red}{\\\\int \\\\texttt{sin}^2(u)\\\\texttt{cos}(u)du}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Simplify]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><p>\\u00a0<\\\/p><p><span style=\\\"font-size: 18.6667px;\\\">Use a U-sub for the third integral. Let \\\\(v=\\\\texttt{sin}(u)\\\\), so \\\\(dv=\\\\texttt{cos}(u)du\\\\):<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\color{red}{\\\\int \\\\texttt{sin}^2(u)\\\\texttt{cos}(u)du}<br \\\/>&amp;= \\\\int \\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{sin}^2(u)}}_{\\\\large v^2}} \\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{cos}(u)du}}_{\\\\large dv}}\\u00a0<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\int v^2dv = \\\\frac{1}{3} v^3\\\\:+ C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{3} \\\\texttt{sin}^3(u)\\\\:+ C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back to u]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{red}{\\\\frac{1}{3} \\\\texttt{sin}^3(2x)\\\\:+ C}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back to x]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><p>\\u00a0<\\\/p><p><span style=\\\"font-size: 18.6667px;\\\">Combine previous results into one expression:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\texttt{sin}^2(x)\\\\texttt{cos}^4(x) dx<br \\\/>&amp;= \\\\frac{x}{16}\\\\:- \\\\frac{1}{64}\\\\texttt{sin}(4x)\\\\:+ \\\\frac{1}{16} \\\\color{red}{\\\\int \\\\texttt{sin}^2(u)\\\\texttt{cos}(u)du}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{x}{16}\\\\:- \\\\frac{1}{64}\\\\texttt{sin}(4x)\\\\:+ \\\\frac{1}{16} \\\\left(\\\\color{red}{\\\\frac{1}{3} \\\\texttt{sin}^3(2x)}\\\\right)\\\\:+ C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Rearrange the expression]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{1}{48}\\\\texttt{sin}^3(2x)\\\\:- \\\\frac{1}{64}\\\\texttt{sin}(4x)\\\\:+ \\\\frac{x}{16}\\\\:+ C}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3b852a84 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3b852a84\" 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-44caab94\" data-id=\"44caab94\" 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-1d22ca46 elementor-widget elementor-widget-toggle\" data-id=\"1d22ca46\" 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-4881\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-4881\" 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-4881\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-4881\"><p><span style=\"font-size: 18.6667px;\">Apply the pythagoras theorem: \\(\\texttt{sin}^2(x)+\\texttt{cos}^2(x) = 1\\), so \\(\\texttt{sin}^2(x) = 1\\:- \\texttt{cos}^2(x)\\)<\/span><\/p><p>\u00a0<\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\texttt{sin}^3(x)\\texttt{cos}^4(x) dx = \\int \\texttt{sin}(x)[1\\:- \\texttt{cos}^2(x)]\\texttt{cos}^4(x)dx<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><p>\u00a0<\/p><p><span style=\"font-size: 18.6667px;\">Apply a U-Sub. Let \\(u=\\texttt{cos}(x)\\), so \\(du=-\\texttt{sin}(x)dx\\). Substituting these values leads to:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\texttt{sin}(x)[1\\:- \\texttt{cos}^2(x)]\\texttt{cos}^4(x)dx<br \/>&amp;= \\int \\color{red}{\\underbrace{\\color{black}{\\texttt{cos}^4}}_{\\large u^4}} \\left[ 1-\\color{red}{\\underbrace{\\color{black}{\\texttt{cos}^2(x)dx}}_{\\large u^2}} \\right] \\color{red}{\\underbrace{\\color{black}{\\texttt{sin}(x)dx}}_{\\large -du}}<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= -\\int u^4(1-u^2)du<br \/>&amp; \\color{gray}{\\text{[Rearrange the expression]}} \\\\[5pt]<br \/>&amp;= -\\int u^4du\\:+ \\int u^6du<br \/>&amp; \\color{gray}{\\text{[Separate the integrals]}} \\\\[5pt]<br \/>&amp;= -\\frac{1}{5}u^5\\:+ \\frac{1}{7}u^7\\:+ C<br \/>&amp; \\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{1}{7}\\texttt{cos}^7(x)\\:- \\frac{1}{5}\\texttt{cos}^5(x)\\:+ C}<br \/>&amp; \\color{gray}{\\text{[Substitute back to x and rearrange]}} \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 2: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Apply the pythagoras theorem: \\\\(\\\\texttt{sin}^2(x)+\\\\texttt{cos}^2(x) = 1\\\\), so \\\\(\\\\texttt{sin}^2(x) = 1\\\\:- \\\\texttt{cos}^2(x)\\\\)<\\\/span><\\\/p><p>\\u00a0<\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\texttt{sin}^3(x)\\\\texttt{cos}^4(x) dx = \\\\int \\\\texttt{sin}(x)[1\\\\:- \\\\texttt{cos}^2(x)]\\\\texttt{cos}^4(x)dx<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><p>\\u00a0<\\\/p><p><span style=\\\"font-size: 18.6667px;\\\">Apply a U-Sub. Let \\\\(u=\\\\texttt{cos}(x)\\\\), so \\\\(du=-\\\\texttt{sin}(x)dx\\\\). Substituting these values leads to:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\texttt{sin}(x)[1\\\\:- \\\\texttt{cos}^2(x)]\\\\texttt{cos}^4(x)dx<br \\\/>&amp;= \\\\int \\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{cos}^4}}_{\\\\large u^4}} \\\\left[ 1-\\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{cos}^2(x)dx}}_{\\\\large u^2}} \\\\right] \\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{sin}(x)dx}}_{\\\\large -du}}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= -\\\\int u^4(1-u^2)du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Rearrange the expression]}} \\\\\\\\[5pt]<br \\\/>&amp;= -\\\\int u^4du\\\\:+ \\\\int u^6du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Separate the integrals]}} \\\\\\\\[5pt]<br \\\/>&amp;= -\\\\frac{1}{5}u^5\\\\:+ \\\\frac{1}{7}u^7\\\\:+ C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{1}{7}\\\\texttt{cos}^7(x)\\\\:- \\\\frac{1}{5}\\\\texttt{cos}^5(x)\\\\:+ C}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back to x and rearrange]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p>\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\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-500befc4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"500befc4\" 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-623ca1a3\" data-id=\"623ca1a3\" 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-354c4172 elementor-widget elementor-widget-toggle\" data-id=\"354c4172\" 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-8941\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-8941\" 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-8941\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-8941\"><p><span style=\"font-size: 18.6667px;\">Apply the pythagoras theorem: \\(\\texttt{sin}^2(x)+\\texttt{cos}^2(x) = 1\\), so \\(\\texttt{sin}^2(x) = 1\\:- \\texttt{cos}^2(x)\\)<\/span><\/p><p>\u00a0<\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\texttt{sin}^3(x)dx = \\int \\texttt{sin}(x) \\cdot \\texttt{sin}^2(x) = \\int \\texttt{sin}(x)[1\\:- \\texttt{cos}^2(x)]dx<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><p>\u00a0<\/p><p><span style=\"font-size: 18.6667px;\">Apply a U-Sub. Let \\(u=\\texttt{cos}(x)\\), so \\(du=-\\texttt{sin}(x)dx\\). Substituting these values leads to:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int [1-\\texttt{cos}^2(x)]\\texttt{sin}(x)dx<br \/>&amp;= \\int [1-\\color{red}{\\underbrace{\\color{black}{\\texttt{cos}^2(x)}}_{\\large u^2}}] \\color{red}{\\underbrace{\\color{black}{\\texttt{sin}(x)}}_{\\large -du}}<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= -\\int (1-u^2)du<br \/>&amp; \\color{gray}{\\text{[Rearrange the expression]}} \\\\[5pt]<br \/>&amp;= -\\int du\\:+ \\int u^2du<br \/>&amp; \\color{gray}{\\text{[Separate the integrals]}} \\\\[5pt]<br \/>&amp;= -u\\:+ \\frac{1}{3}u^3\\:+ C<br \/>&amp; \\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{1}{3}\\texttt{cos}^3(x)\\:- \\texttt{cos}(x)\\:+ C}<br \/>&amp; \\color{gray}{\\text{[Substitute back to x and rearrange]}} \\\\[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;\\\">Apply the pythagoras theorem: \\\\(\\\\texttt{sin}^2(x)+\\\\texttt{cos}^2(x) = 1\\\\), so \\\\(\\\\texttt{sin}^2(x) = 1\\\\:- \\\\texttt{cos}^2(x)\\\\)<\\\/span><\\\/p><p>\\u00a0<\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\texttt{sin}^3(x)dx = \\\\int \\\\texttt{sin}(x) \\\\cdot \\\\texttt{sin}^2(x) = \\\\int \\\\texttt{sin}(x)[1\\\\:- \\\\texttt{cos}^2(x)]dx<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><p>\\u00a0<\\\/p><p><span style=\\\"font-size: 18.6667px;\\\">Apply a U-Sub. Let \\\\(u=\\\\texttt{cos}(x)\\\\), so \\\\(du=-\\\\texttt{sin}(x)dx\\\\). Substituting these values leads to:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int [1-\\\\texttt{cos}^2(x)]\\\\texttt{sin}(x)dx<br \\\/>&amp;= \\\\int [1-\\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{cos}^2(x)}}_{\\\\large u^2}}] \\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{sin}(x)}}_{\\\\large -du}}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= -\\\\int (1-u^2)du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Rearrange the expression]}} \\\\\\\\[5pt]<br \\\/>&amp;= -\\\\int du\\\\:+ \\\\int u^2du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Separate the integrals]}} \\\\\\\\[5pt]<br \\\/>&amp;= -u\\\\:+ \\\\frac{1}{3}u^3\\\\:+ C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{1}{3}\\\\texttt{cos}^3(x)\\\\:- \\\\texttt{cos}(x)\\\\:+ C}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back to x and rearrange]}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-82e1d91 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"82e1d91\" 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-794d0aca\" data-id=\"794d0aca\" 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-23bc1874 elementor-widget elementor-widget-toggle\" data-id=\"23bc1874\" 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-5991\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-5991\" 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-5991\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-5991\"><p><span style=\"font-size: 18.6667px;\">Apply the sine and cosine product identity: \\(\\texttt{sin}(a)\\texttt{cos}(b) = \\frac{1}{2} \\left( \\texttt{sin}(a+b) + \\texttt{sin}(a-b) \\right)\\)<\/span><\/p><p>\u00a0<\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int^{\\pi}_{-\\pi} \\texttt{sin}(3x)\\texttt{cos}(2x)dx<br \/>&amp;= \\frac{1}{2}\\int^{\\pi}_{-\\pi} [\\texttt{sin}(5x) + \\texttt{sin}(x)]dx<br \/>&amp; \\color{gray}{\\text{[Apply the identity]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\left(\\int^{\\pi}_{-\\pi} \\texttt{sin}(5x)dx\\:+ \\int^{\\pi}_{-\\pi} \\texttt{sin}(x)dx\u00a0\\right)<br \/>&amp; \\color{gray}{\\text{[Separate the integrand]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\left[ -\\frac{1}{5}\\texttt{cos}(5x)\\:- \\texttt{cos}(x) \\right] \\Bigr|^{\\pi}_{-\\pi}<br \/>&amp; \\color{gray}{\\text{[Integrate using a quick U-sub]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\left( \\left[ -\\frac{1}{5}\\texttt{cos}(5\\pi)\\:- \\texttt{cos}(\\pi) \\right]\\:-\u00a0\\left[ -\\frac{1}{5}\\texttt{cos}(-5\\pi)\\:- \\texttt{cos}(-\\pi)\\right] \\right)<br \/>&amp; \\color{gray}{\\text{[Evaluate the antiderivative]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\left( \\left[ -\\frac{1}{5}(-1)\\:- (-1) \\right]\\:-\u00a0\\left[ -\\frac{1}{5}(-1)\\:- (-1)\\right] \\right) \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\left( \\left[ \\frac{1}{5}\\:+ 1 \\right]\\:-\u00a0\\left[ \\frac{1}{5}\\:+ 1\\right] \\right) \\\\[5pt]<br \/>&amp;= \\frac{1}{2} \\left( \\frac{6}{5}\\:- \\frac{6}{5} \\right) \\\\[5pt]<br \/>&amp;= \\frac{1}{2} (0) \\\\[5pt]<br \/>&amp;= \\color{green}{0}<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;\\\">Apply the sine and cosine product identity: \\\\(\\\\texttt{sin}(a)\\\\texttt{cos}(b) = \\\\frac{1}{2} \\\\left( \\\\texttt{sin}(a+b) + \\\\texttt{sin}(a-b) \\\\right)\\\\)<\\\/span><\\\/p><p>\\u00a0<\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int^{\\\\pi}_{-\\\\pi} \\\\texttt{sin}(3x)\\\\texttt{cos}(2x)dx<br \\\/>&amp;= \\\\frac{1}{2}\\\\int^{\\\\pi}_{-\\\\pi} [\\\\texttt{sin}(5x) + \\\\texttt{sin}(x)]dx<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the identity]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\left(\\\\int^{\\\\pi}_{-\\\\pi} \\\\texttt{sin}(5x)dx\\\\:+ \\\\int^{\\\\pi}_{-\\\\pi} \\\\texttt{sin}(x)dx\\u00a0\\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Separate the integrand]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\left[ -\\\\frac{1}{5}\\\\texttt{cos}(5x)\\\\:- \\\\texttt{cos}(x) \\\\right] \\\\Bigr|^{\\\\pi}_{-\\\\pi}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate using a quick U-sub]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\left( \\\\left[ -\\\\frac{1}{5}\\\\texttt{cos}(5\\\\pi)\\\\:- \\\\texttt{cos}(\\\\pi) \\\\right]\\\\:-\\u00a0\\\\left[ -\\\\frac{1}{5}\\\\texttt{cos}(-5\\\\pi)\\\\:- \\\\texttt{cos}(-\\\\pi)\\\\right] \\\\right)<br \\\/>&amp; \\\\color{gray}{\\\\text{[Evaluate the antiderivative]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\left( \\\\left[ -\\\\frac{1}{5}(-1)\\\\:- (-1) \\\\right]\\\\:-\\u00a0\\\\left[ -\\\\frac{1}{5}(-1)\\\\:- (-1)\\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\left( \\\\left[ \\\\frac{1}{5}\\\\:+ 1 \\\\right]\\\\:-\\u00a0\\\\left[ \\\\frac{1}{5}\\\\:+ 1\\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} \\\\left( \\\\frac{6}{5}\\\\:- \\\\frac{6}{5} \\\\right) \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{2} (0) \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{0}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-34b53812 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"34b53812\" 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-10e00430\" data-id=\"10e00430\" 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-1f38ce57 elementor-widget elementor-widget-toggle\" data-id=\"1f38ce57\" 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-5231\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-5231\" 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-5231\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-5231\"><p><span style=\"font-size: 18.6667px;\">Apply the pythagoras theorem: \\(\\texttt{sin}^2(x)+\\texttt{cos}^2(x) = 1\\), so \\(\\texttt{sin}^2(x) = 1\\:- \\texttt{cos}^2(x)\\)<\/span><\/p><p>\u00a0<\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int^{\\frac{\\pi}{4}}_0 \\texttt{sin}^3(x)dx = \\int^{\\frac{\\pi}{4}}_0 \\texttt{sin}(x) \\cdot \\texttt{sin}^2(x) = \\int^{\\frac{\\pi}{4}}_0 \\texttt{sin}(x)[1\\:- \\texttt{cos}^2(x)]dx<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><p>\u00a0<\/p><p><span style=\"font-size: 18.6667px;\">Apply a U-Sub. Let \\(u=\\texttt{cos}(x)\\), so \\(du=-\\texttt{sin}(x)dx\\). Substituting these values leads to:<\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int^{\\frac{\\pi}{4}}_0 [1-\\texttt{cos}^2(x)]\\texttt{sin}(x)dx<br \/>&amp;= \\int^{\\frac{\\pi}{4}}_0 [1-\\color{red}{\\underbrace{\\color{black}{\\texttt{cos}^2(x)}}_{\\large u^2}}] \\color{red}{\\underbrace{\\color{black}{\\texttt{sin}(x)}}_{\\large -du}}<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= -\\int^{\\frac{\\pi}{4}}_0 (1-u^2)du<br \/>&amp; \\color{gray}{\\text{[Rearrange the expression]}} \\\\[5pt]<br \/>&amp;= -\\int^{\\frac{\\pi}{4}}_0 du\\:+ \\int^{\\frac{\\pi}{4}}_0 u^2du<br \/>&amp; \\color{gray}{\\text{[Separate the integrals]}} \\\\[5pt]<br \/>&amp;= [-u\\:+ \\frac{1}{3}u^3]\\Big|^{\\frac{\\pi}{4}}_0<br \/>&amp; \\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\left[ \\frac{1}{3}\\texttt{cos}^3(x)\\:- \\texttt{cos}(x) \\right] \\Big|^{\\frac{\\pi}{4}}_0<br \/>&amp; \\color{gray}{\\text{[Substitute back to x and rearrange]}} \\\\[5pt]<br \/>&amp;= \\left[ \\frac{1}{3}\\texttt{cos}^3(\\frac{\\pi}{4})\\:- \\texttt{cos}(\\frac{\\pi}{4}) \\right]\\:- \\left[ \\frac{1}{3}\\texttt{cos}^3(0)\\:- \\texttt{cos}(0) \\right]<br \/>&amp; \\color{gray}{\\text{[Evaluate the antiderivative]}} \\\\[5pt]<br \/>&amp;= \\left[ \\frac{1}{3}\\left( \\frac{\\sqrt{2}}{2} \\right)^3\\:- \\frac{\\sqrt{2}}{2}\u00a0\\right]\\:- \\left[ \\frac{1}{3}(1)^3\\:- 1 \\right] \\\\[5pt]<br \/>&amp;= \\left[ \\frac{1}{3}\\left( \\frac{2\\sqrt{2}}{8} \\right)\\:- \\frac{\\sqrt{2}}{2}\u00a0\\right]\\:+\\frac{2}{3} \\\\[5pt]<br \/>&amp;= \\frac{\\sqrt{2}}{12}\\:- \\frac{\\sqrt{2}}{2}\\:+\\frac{2}{3} \\\\[5pt]<br \/>&amp;= -\\frac{5\\sqrt{2}}{12}\\:+ \\frac{2}{3} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{2}{3}\\:- \\frac{5\\sqrt{2}}{12}} \\\\[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;\\\">Apply the pythagoras theorem: \\\\(\\\\texttt{sin}^2(x)+\\\\texttt{cos}^2(x) = 1\\\\), so \\\\(\\\\texttt{sin}^2(x) = 1\\\\:- \\\\texttt{cos}^2(x)\\\\)<\\\/span><\\\/p><p>\\u00a0<\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int^{\\\\frac{\\\\pi}{4}}_0 \\\\texttt{sin}^3(x)dx = \\\\int^{\\\\frac{\\\\pi}{4}}_0 \\\\texttt{sin}(x) \\\\cdot \\\\texttt{sin}^2(x) = \\\\int^{\\\\frac{\\\\pi}{4}}_0 \\\\texttt{sin}(x)[1\\\\:- \\\\texttt{cos}^2(x)]dx<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><p>\\u00a0<\\\/p><p><span style=\\\"font-size: 18.6667px;\\\">Apply a U-Sub. Let \\\\(u=\\\\texttt{cos}(x)\\\\), so \\\\(du=-\\\\texttt{sin}(x)dx\\\\). Substituting these values leads to:<\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int^{\\\\frac{\\\\pi}{4}}_0 [1-\\\\texttt{cos}^2(x)]\\\\texttt{sin}(x)dx<br \\\/>&amp;= \\\\int^{\\\\frac{\\\\pi}{4}}_0 [1-\\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{cos}^2(x)}}_{\\\\large u^2}}] \\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{sin}(x)}}_{\\\\large -du}}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= -\\\\int^{\\\\frac{\\\\pi}{4}}_0 (1-u^2)du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Rearrange the expression]}} \\\\\\\\[5pt]<br \\\/>&amp;= -\\\\int^{\\\\frac{\\\\pi}{4}}_0 du\\\\:+ \\\\int^{\\\\frac{\\\\pi}{4}}_0 u^2du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Separate the integrals]}} \\\\\\\\[5pt]<br \\\/>&amp;= [-u\\\\:+ \\\\frac{1}{3}u^3]\\\\Big|^{\\\\frac{\\\\pi}{4}}_0<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\left[ \\\\frac{1}{3}\\\\texttt{cos}^3(x)\\\\:- \\\\texttt{cos}(x) \\\\right] \\\\Big|^{\\\\frac{\\\\pi}{4}}_0<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back to x and rearrange]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\left[ \\\\frac{1}{3}\\\\texttt{cos}^3(\\\\frac{\\\\pi}{4})\\\\:- \\\\texttt{cos}(\\\\frac{\\\\pi}{4}) \\\\right]\\\\:- \\\\left[ \\\\frac{1}{3}\\\\texttt{cos}^3(0)\\\\:- \\\\texttt{cos}(0) \\\\right]<br \\\/>&amp; \\\\color{gray}{\\\\text{[Evaluate the antiderivative]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\left[ \\\\frac{1}{3}\\\\left( \\\\frac{\\\\sqrt{2}}{2} \\\\right)^3\\\\:- \\\\frac{\\\\sqrt{2}}{2}\\u00a0\\\\right]\\\\:- \\\\left[ \\\\frac{1}{3}(1)^3\\\\:- 1 \\\\right] \\\\\\\\[5pt]<br \\\/>&amp;= \\\\left[ \\\\frac{1}{3}\\\\left( \\\\frac{2\\\\sqrt{2}}{8} \\\\right)\\\\:- \\\\frac{\\\\sqrt{2}}{2}\\u00a0\\\\right]\\\\:+\\\\frac{2}{3} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{\\\\sqrt{2}}{12}\\\\:- \\\\frac{\\\\sqrt{2}}{2}\\\\:+\\\\frac{2}{3} \\\\\\\\[5pt]<br \\\/>&amp;= -\\\\frac{5\\\\sqrt{2}}{12}\\\\:+ \\\\frac{2}{3} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{2}{3}\\\\:- \\\\frac{5\\\\sqrt{2}}{12}} \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-262c794e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"262c794e\" 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-5f1ade5a\" data-id=\"5f1ade5a\" 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-19bfb59a elementor-widget elementor-widget-toggle\" data-id=\"19bfb59a\" 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-4311\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-4311\" 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-4311\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-4311\"><p><span style=\"font-size: 18.6667px;\">Apply a quick U-sub. Let \\(u=4x\\), so \\(du=4dx\\). Substituting these values yield:<\/span><\/p><p>\u00a0<\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\texttt{sec}(4x)dx<br \/>&amp;= \\int \\texttt{sec}(\\color{red}{\\underbrace{\\color{black}{4x}}_{\\large u}}) \\color{red}{\\underbrace{\\color{black}{dx}}_{\\large \\frac{du}{4}}}<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{4}\\int \\texttt{sec}(u)du<br \/>&amp; \\color{gray}{\\text{[Rearrange the expression]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{4}\\texttt{ln} |\\texttt{sec}(u)\\:+ \\texttt{tan}(u)| \\:+ C<br \/>&amp; \\color{gray}{\\text{[Integrate]}} \\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{1}{4}\\texttt{ln}|\\texttt{sec}(4x)\\:+ \\texttt{tan}(4x)|\\:+ 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 6: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-size: 18.6667px;\\\">Apply a quick U-sub. Let \\\\(u=4x\\\\), so \\\\(du=4dx\\\\). Substituting these values yield:<\\\/span><\\\/p><p>\\u00a0<\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\texttt{sec}(4x)dx<br \\\/>&amp;= \\\\int \\\\texttt{sec}(\\\\color{red}{\\\\underbrace{\\\\color{black}{4x}}_{\\\\large u}}) \\\\color{red}{\\\\underbrace{\\\\color{black}{dx}}_{\\\\large \\\\frac{du}{4}}}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{4}\\\\int \\\\texttt{sec}(u)du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Rearrange the expression]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{4}\\\\texttt{ln} |\\\\texttt{sec}(u)\\\\:+ \\\\texttt{tan}(u)| \\\\:+ C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{1}{4}\\\\texttt{ln}|\\\\texttt{sec}(4x)\\\\:+ \\\\texttt{tan}(4x)|\\\\:+ 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\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-793f3d82 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"793f3d82\" 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-5ed79169\" data-id=\"5ed79169\" 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-18fc15ed elementor-widget elementor-widget-toggle\" data-id=\"18fc15ed\" 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-4191\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-4191\" 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-4191\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-4191\"><p><span style=\"font-size: 18.6667px;\">Apply a quick U-sub. Let \\(u=3x\\), so \\(du=3dx\\). Substituting these values yield:<br \/><\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\int \\texttt{tan}^3(3x) \\texttt{sec}^2(3x) dx<br \/>&amp;= \\int \\texttt{tan}^3 \\color{red}{\\underbrace{\\color{black}{(3x)}}_{\\large u}}\u00a0\\texttt{sec}^2 \\color{red}{\\underbrace{\\color{black}{(3x)}}_{\\large u}} \\color{red}{\\underbrace{\\color{black}{dx}}_{\\large \\frac{du}{3}}}<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{3}\\int \\texttt{tan}^3(u) \\texttt{sec}^2(u)du<br \/>&amp; \\color{gray}{\\text{[Simplify]}}\u00a0\\\\[5pt]<br \/>&amp;= \\frac{1}{3}\\int \\texttt{tan}(u) \\color{red}{\\underbrace{\\color{black}{[\\texttt{sec}^2(u)\\:- 1]}}_{\\texttt{tan}^2(u)}} \\texttt{sec}^2(u)du<br \/>&amp; \\color{gray}{\\text{[Apply Pythagoras Theorem]}}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><p><span style=\"font-size: 18.6667px;\">Apply a quick U-sub. Let \\(v=\\texttt{sec}(u)\\), so \\(dv=\\texttt{sec}(u)\\texttt{tan}(u)du\\). Substituting these values yield:<br \/><\/span><\/p>\\(\\normalsize<br \/>\\begin{align*}\\displaystyle<br \/>\\frac{1}{3} \\int [\\texttt{sec}^2(u)\\:- 1]\\texttt{sec}^2(u)\\texttt{tan}(u)du<br \/>&amp;= \\frac{1}{3} \\int [\\color{red}{\\underbrace{\\color{black}{\\texttt{sec}^2(u)}}_{\\large v^2}}\\:- 1] [\\color{red}{\\underbrace{\\color{black}{\\texttt{sec}(u)}}_{\\large v}}] [\\color{red}{\\underbrace{\\color{black}{\\texttt{sec}(u)\\texttt{tan}(u)du}}_{\\large dv}}]<br \/>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br \/>&amp;= \\frac{1}{3}\\int (v^2\\:- 1)vdv<br \/>&amp; \\color{gray}{\\text{[Simplify]}}\u00a0\\\\[5pt]<br \/>&amp;= \\frac{1}{3}\\int (v^3\\:- v)dv \\\\[5pt]<br \/>&amp;= \\frac{1}{3}\\left( \\frac{1}{4}v^4\\:- \\frac{1}{2}v^2 \\right)\\:+ C<br \/>&amp; \\color{gray}{\\text{[Integrate]}}\u00a0\\\\[5pt]<br \/>&amp;= \\frac{1}{12}v^4\\:- \\frac{1}{6}v^2\\:+ C<br \/>&amp; \\color{gray}{\\text{[Simplify]}}\u00a0\\\\[5pt]<br \/>&amp;= \\frac{1}{12}\\texttt{sec}^4(u)\\:- \\frac{1}{6}\\texttt{sec}^2(u)\\:+ C<br \/>&amp; \\color{gray}{\\text{[Substitute back to u]}}\u00a0\\\\[5pt]<br \/>&amp;= \\color{green}{\\frac{1}{12}\\texttt{sec}^4(3x)\\:- \\frac{1}{6}\\texttt{sec}^2(3x)\\:+ C}<br \/>&amp; \\color{gray}{\\text{[Substitute back to x]}}\u00a0\\\\[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;\\\">Apply a quick U-sub. Let \\\\(u=3x\\\\), so \\\\(du=3dx\\\\). Substituting these values yield:<br \\\/><\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\int \\\\texttt{tan}^3(3x) \\\\texttt{sec}^2(3x) dx<br \\\/>&amp;= \\\\int \\\\texttt{tan}^3 \\\\color{red}{\\\\underbrace{\\\\color{black}{(3x)}}_{\\\\large u}}\\u00a0\\\\texttt{sec}^2 \\\\color{red}{\\\\underbrace{\\\\color{black}{(3x)}}_{\\\\large u}} \\\\color{red}{\\\\underbrace{\\\\color{black}{dx}}_{\\\\large \\\\frac{du}{3}}}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{3}\\\\int \\\\texttt{tan}^3(u) \\\\texttt{sec}^2(u)du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Simplify]}}\\u00a0\\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{3}\\\\int \\\\texttt{tan}(u) \\\\color{red}{\\\\underbrace{\\\\color{black}{[\\\\texttt{sec}^2(u)\\\\:- 1]}}_{\\\\texttt{tan}^2(u)}} \\\\texttt{sec}^2(u)du<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply Pythagoras Theorem]}}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><p><span style=\\\"font-size: 18.6667px;\\\">Apply a quick U-sub. Let \\\\(v=\\\\texttt{sec}(u)\\\\), so \\\\(dv=\\\\texttt{sec}(u)\\\\texttt{tan}(u)du\\\\). Substituting these values yield:<br \\\/><\\\/span><\\\/p>\\\\(\\\\normalsize<br \\\/>\\\\begin{align*}\\\\displaystyle<br \\\/>\\\\frac{1}{3} \\\\int [\\\\texttt{sec}^2(u)\\\\:- 1]\\\\texttt{sec}^2(u)\\\\texttt{tan}(u)du<br \\\/>&amp;= \\\\frac{1}{3} \\\\int [\\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{sec}^2(u)}}_{\\\\large v^2}}\\\\:- 1] [\\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{sec}(u)}}_{\\\\large v}}] [\\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{sec}(u)\\\\texttt{tan}(u)du}}_{\\\\large dv}}]<br \\\/>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{3}\\\\int (v^2\\\\:- 1)vdv<br \\\/>&amp; \\\\color{gray}{\\\\text{[Simplify]}}\\u00a0\\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{3}\\\\int (v^3\\\\:- v)dv \\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{3}\\\\left( \\\\frac{1}{4}v^4\\\\:- \\\\frac{1}{2}v^2 \\\\right)\\\\:+ C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Integrate]}}\\u00a0\\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{12}v^4\\\\:- \\\\frac{1}{6}v^2\\\\:+ C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Simplify]}}\\u00a0\\\\\\\\[5pt]<br \\\/>&amp;= \\\\frac{1}{12}\\\\texttt{sec}^4(u)\\\\:- \\\\frac{1}{6}\\\\texttt{sec}^2(u)\\\\:+ C<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back to u]}}\\u00a0\\\\\\\\[5pt]<br \\\/>&amp;= \\\\color{green}{\\\\frac{1}{12}\\\\texttt{sec}^4(3x)\\\\:- \\\\frac{1}{6}\\\\texttt{sec}^2(3x)\\\\:+ C}<br \\\/>&amp; \\\\color{gray}{\\\\text{[Substitute back to x]}}\\u00a0\\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7c757776 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7c757776\" 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-52186b\" data-id=\"52186b\" 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-4ac0b2fb elementor-widget elementor-widget-toggle\" data-id=\"4ac0b2fb\" 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-1251\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1251\" 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-1251\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1251\"><p><span style=\"font-size: 18.6667px;\">Apply a quick U-sub. Let \\(u=\\texttt{tan}(x)\\), so \\(du=\\texttt{sec}^2(x)dx\\). Substituting these values yield:<br><\/span><\/p>\n\\(\\normalsize<br>\\begin{align*}\\displaystyle<br>\\int \\texttt{tan}^4(x) \\texttt{sec}^2(x) dx<br>&amp;= \\int \\color{red}{\\underbrace{\\color{black}{\\texttt{tan}^4(x)}}_{\\large u^4}} \\color{red}{\\underbrace{\\color{black}{\\texttt{sec}^2(x)dx}}_{\\large du}}<br>&amp; \\color{gray}{\\text{[Apply the substitution]}} \\\\[5pt]<br>&amp;= \\int u^4du<br>&amp; \\color{gray}{\\text{[Simplify]}}&nbsp;\\\\[5pt]<br>&amp;= \\frac{1}{5} u^5\\:+ C<br>&amp; \\color{gray}{\\text{[Integrate]}}&nbsp;\\\\[5pt]<br>&amp;= \\color{green}{\\frac{1}{5} \\texttt{tan}^5(x)\\:+ C}<br>&amp; \\color{gray}{\\text{[Substitute back to x]}}<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;\\\">Apply a quick U-sub. Let \\\\(u=\\\\texttt{tan}(x)\\\\), so \\\\(du=\\\\texttt{sec}^2(x)dx\\\\). Substituting these values yield:<br><\\\/span><\\\/p>\\n\\\\(\\\\normalsize<br>\\\\begin{align*}\\\\displaystyle<br>\\\\int \\\\texttt{tan}^4(x) \\\\texttt{sec}^2(x) dx<br>&amp;= \\\\int \\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{tan}^4(x)}}_{\\\\large u^4}} \\\\color{red}{\\\\underbrace{\\\\color{black}{\\\\texttt{sec}^2(x)dx}}_{\\\\large du}}<br>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}} \\\\\\\\[5pt]<br>&amp;= \\\\int u^4du<br>&amp; \\\\color{gray}{\\\\text{[Simplify]}}&nbsp;\\\\\\\\[5pt]<br>&amp;= \\\\frac{1}{5} u^5\\\\:+ C<br>&amp; \\\\color{gray}{\\\\text{[Integrate]}}&nbsp;\\\\\\\\[5pt]<br>&amp;= \\\\color{green}{\\\\frac{1}{5} \\\\texttt{tan}^5(x)\\\\:+ C}<br>&amp; \\\\color{gray}{\\\\text{[Substitute back to x]}}<br>\\\\end{align*}<br>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ad013a0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ad013a0\" 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-66404e1a\" data-id=\"66404e1a\" 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-dd5165d elementor-widget elementor-widget-toggle\" data-id=\"dd5165d\" 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-2321\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2321\" 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-2321\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2321\"><p><span style=\"font-size: 18.6667px;\">Use the power reduction identity in order to transform the integrand by parts, first the numerator and then the denominator. Let \\(\\texttt{sin}^2(x)=\\frac{1}{2}[1-\\texttt{cos}(2x)]\\), so \\(2\\texttt{sin}^2(x)=1-\\texttt{cos}(2x)\\). Likewise, let&nbsp;<\/span><span style=\"font-size: 18.6667px;\">\\(\\texttt{cos}^2(x)=\\frac{1}{2}[1+\\texttt{cos}(2x)]\\), so \\(2\\texttt{cos}^2(x)=1+\\texttt{cos}(2x)\\). Substituting these new values in the integrand yields:<\/span><span style=\"font-size: 18.6667px;\"><br><\/span><\/p>\n\\(\\normalsize<br>\\begin{align*}\\displaystyle<br>\\int \\frac{1+\\texttt{cos}(2x)}{1-\\texttt{cos}(2x)}dx<br>&amp;= \\int \\frac{\\color{red}{2\\texttt{cos}^2(x)}}{\\color{red}{2\\texttt{sin}^2(x)}} dx<br>&amp; \\color{gray}{\\text{[Apply the substitution]}}&nbsp;\\\\[5pt]<br>&amp;= \\int \\texttt{cot}^2(x)dx<br>&amp; \\color{gray}{\\text{[Simplify and use trigonometric identitites]}}&nbsp;\\\\[5pt]<br>&amp;= \\int [\\texttt{cot}^2(x)+1-1]dx<br>&amp; \\color{gray}{\\text{[Add zero to the integrand]}}&nbsp;\\\\[5pt]<br>&amp;= \\int [\\texttt{cot}^2(x)+1]dx\\:- \\int dx<br>&amp; \\color{gray}{\\text{[Separate the integrals]}}&nbsp;\\\\[5pt]<br>&amp;= \\int \\texttt{csc}^2(x)dx\\:- \\int dx<br>&amp; \\color{gray}{\\text{[Simplify using trigonometric identitites]}}&nbsp;\\\\[5pt]<br>&amp;= \\color{green}{\\texttt{cot}(x)\\:-x\\:+C}<br>&amp; \\color{gray}{\\text{[Integrate]}}&nbsp;\\\\[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;\\\">Use the power reduction identity in order to transform the integrand by parts, first the numerator and then the denominator. Let \\\\(\\\\texttt{sin}^2(x)=\\\\frac{1}{2}[1-\\\\texttt{cos}(2x)]\\\\), so \\\\(2\\\\texttt{sin}^2(x)=1-\\\\texttt{cos}(2x)\\\\). Likewise, let&nbsp;<\\\/span><span style=\\\"font-size: 18.6667px;\\\">\\\\(\\\\texttt{cos}^2(x)=\\\\frac{1}{2}[1+\\\\texttt{cos}(2x)]\\\\), so \\\\(2\\\\texttt{cos}^2(x)=1+\\\\texttt{cos}(2x)\\\\). Substituting these new values in the integrand yields:<\\\/span><span style=\\\"font-size: 18.6667px;\\\"><br><\\\/span><\\\/p>\\n\\\\(\\\\normalsize<br>\\\\begin{align*}\\\\displaystyle<br>\\\\int \\\\frac{1+\\\\texttt{cos}(2x)}{1-\\\\texttt{cos}(2x)}dx<br>&amp;= \\\\int \\\\frac{\\\\color{red}{2\\\\texttt{cos}^2(x)}}{\\\\color{red}{2\\\\texttt{sin}^2(x)}} dx<br>&amp; \\\\color{gray}{\\\\text{[Apply the substitution]}}&nbsp;\\\\\\\\[5pt]<br>&amp;= \\\\int \\\\texttt{cot}^2(x)dx<br>&amp; \\\\color{gray}{\\\\text{[Simplify and use trigonometric identitites]}}&nbsp;\\\\\\\\[5pt]<br>&amp;= \\\\int [\\\\texttt{cot}^2(x)+1-1]dx<br>&amp; \\\\color{gray}{\\\\text{[Add zero to the integrand]}}&nbsp;\\\\\\\\[5pt]<br>&amp;= \\\\int [\\\\texttt{cot}^2(x)+1]dx\\\\:- \\\\int dx<br>&amp; \\\\color{gray}{\\\\text{[Separate the integrals]}}&nbsp;\\\\\\\\[5pt]<br>&amp;= \\\\int \\\\texttt{csc}^2(x)dx\\\\:- \\\\int dx<br>&amp; \\\\color{gray}{\\\\text{[Simplify using trigonometric identitites]}}&nbsp;\\\\\\\\[5pt]<br>&amp;= \\\\color{green}{\\\\texttt{cot}(x)\\\\:-x\\\\:+C}<br>&amp; \\\\color{gray}{\\\\text{[Integrate]}}&nbsp;\\\\\\\\[5pt]<br>\\\\end{align*}<br>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-66de00a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"66de00a\" 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-14a68262\" data-id=\"14a68262\" 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-53d0087f elementor-widget elementor-widget-toggle\" data-id=\"53d0087f\" 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-1401\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1401\" 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-1401\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1401\"><p><span style=\"font-size: 18.6667px;\">Multiply the integrand by its reciprocal in a fraction:<br><\/span><\/p>\n\\(\\normalsize<br>\\begin{align*}\\displaystyle<br>\\int \\sqrt{1-\\texttt{cos}(\\theta)} d\\theta<br>&amp;= \\int \\sqrt{1-\\texttt{cos}(\\theta)} \\left( \\frac{\\sqrt{1+\\texttt{cos}(\\theta)}}{\\sqrt{1+\\texttt{cos}(\\theta)}} \\right) d\\theta \\\\[5pt]<br>&amp;= \\int \\frac{\\sqrt{1-\\texttt{cos}^2(\\theta)}}{\\sqrt{1+\\texttt{cos}(\\theta)}} d\\theta<br>&amp; \\color{gray}{\\text{[Simplify]}}&nbsp;\\\\[5pt]<br>&amp;= \\int \\frac{\\sqrt{\\texttt{sin}^2(\\theta)}}{\\sqrt{1+\\texttt{cos}(\\theta)}} d\\theta&nbsp;\\\\[5pt]<br>&amp;= \\int \\frac{\\texttt{sin}(\\theta)}{\\sqrt{1+\\texttt{cos}(\\theta)}} d\\theta&nbsp;\\\\[5pt]<br>\\end{align*}<br>\\)\n<p><br><\/p>\n<p><\/p>\n<p><span style=\"font-size: 18.6667px;\">Perform a quick U-Sub. Let \\(u=1\\:+ \\texttt{cos}(\\theta)\\), so \\(du=-\\texttt{sin}(\\theta)d\\theta\\):<br><\/span><\/p>\n\\(\\normalsize<br>\\begin{align*}\\displaystyle<br>\\int \\frac{\\texttt{sin}(\\theta)}{\\sqrt{1+\\texttt{cos}(\\theta)}} d\\theta<br>&amp;= \\int \\frac{\\color{red}{-du}}{\\color{red}{\\sqrt{u}}}<br>&amp; \\color{gray}{\\text{[Apply the substitution]}}&nbsp;\\\\[5pt]<br>&amp;= -2\\sqrt{u}\\:+ C<br>&amp; \\color{gray}{\\text{[Integrate]}}&nbsp;\\\\[5pt]<br>&amp;= \\color{green}{-2\\sqrt{1\\:+ \\texttt{cos}(\\theta)}\\:+ C}<br>&amp; \\color{gray}{\\text{[Substitute back for \\(\\theta\\)]}}&nbsp;\\\\[5pt]<br>\\end{align*}<br>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ccde6a5 elementor-section-stretched elementor-section-boxed elementor-section-height-default 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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 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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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menu-item-9815\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-techinques\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Integration Techinques<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-13700\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-applications\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Integration Applications<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19550\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/\" class=\"elementor-sub-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14330\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Sequences and Series<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11364\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-9804\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/convergence-test\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Convergence Test<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10233\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/power-series\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Power Series<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11363\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/taylor-series\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Taylor Series<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-21607\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/problem-set-sequences-series\/\" class=\"elementor-sub-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14331\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Parametric Equations<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12174\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/fundamentals-of-parametric-equations\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19376\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/polar-coordinates-calculus\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Polar Coordinates<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11366\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/precalculus-resources\/unit-circle\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Unit Circle<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12198\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/calculus-parametric-equation\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Calculus of Parametric Equations<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-21606\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/problem-set-parametric-eqs-2\/\" class=\"elementor-sub-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12615\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/virtual-lessons-parametric-equations\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10232\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/ap-calculus-frqs\/\" class=\"elementor-item\" tabindex=\"-1\">AP Calculus Prep<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-1c1e1cd\" data-id=\"1c1e1cd\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-788e7a3 elementor-widget elementor-widget-spacer\" data-id=\"788e7a3\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Practice Problems: Trigonometric Integrals When integrating products of trigonometric functions, the general practice involves applying the trigonometric versions of the Pythagorean Theorem such as\u00a0 \u00a0 \u00a0 or\u00a0 \u00a0 in conjunction with an appropriate u-substitution. 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