{"id":668,"date":"2021-05-10T18:52:11","date_gmt":"2021-05-10T18:52:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=668"},"modified":"2021-11-17T02:25:03","modified_gmt":"2021-11-17T02:25:03","slug":"summary-of-trigonometric-integrals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-trigonometric-integrals\/","title":{"raw":"Summary of Trigonometric Integrals","rendered":"Summary of Trigonometric Integrals"},"content":{"raw":"<section id=\"fs-id1165043207351\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1165043207359\" data-bullet-style=\"bullet\">\r\n \t<li>Integrals of trigonometric functions can be evaluated by the use of various strategies. These strategies include:\r\n<ol id=\"fs-id1165043207368\" type=\"1\">\r\n \t<li>Applying trigonometric identities to rewrite the integral so that it may be evaluated by <em data-effect=\"italics\">u<\/em>-substitution<\/li>\r\n \t<li>Using integration by parts<\/li>\r\n \t<li>Applying trigonometric identities to rewrite products of sines and cosines with different arguments as the sum of individual sine and cosine functions<\/li>\r\n \t<li>Applying reduction formulas<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165043207398\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<p id=\"fs-id1165039565319\">To integrate products involving [latex]\\sin\\left(ax\\right)[\/latex], [latex]\\sin\\left(bx\\right)[\/latex], [latex]\\cos\\left(ax\\right)[\/latex], and [latex]\\cos\\left(bx\\right)[\/latex], use the substitutions.<\/p>\r\n\r\n<ul id=\"fs-id1165042558473\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Sine Products<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\sin\\left(ax\\right)\\sin\\left(bx\\right)=\\frac{1}{2}\\cos\\left(\\left(a-b\\right)x\\right)-\\frac{1}{2}\\cos\\left(\\left(a+b\\right)x\\right)[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Sine and Cosine Products<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\sin\\left(ax\\right)\\cos\\left(bx\\right)=\\frac{1}{2}\\sin\\left(\\left(a-b\\right)x\\right)+\\frac{1}{2}\\sin\\left(\\left(a+b\\right)x\\right)[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Cosine Products<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\cos\\left(ax\\right)\\cos\\left(bx\\right)=\\frac{1}{2}\\cos\\left(\\left(a-b\\right)x\\right)+\\frac{1}{2}\\cos\\left(\\left(a+b\\right)x\\right)[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Power Reduction Formula<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]{\\displaystyle\\int}{\\text{sec}}^{n}xdx=\\frac{1}{n - 1}{\\text{sec}}^{n - 1}x+\\frac{n - 2}{n - 1}{\\displaystyle\\int}{\\text{sec}}^{n - 2}xdx[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Power Reduction Formula<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]{\\displaystyle\\int}{\\tan}^{n}xdx=\\frac{1}{n - 1}{\\tan}^{n - 1}x-{\\displaystyle\\int}{\\tan}^{n - 2}xdx[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-id1165042637870\" class=\"section-exercises\" data-depth=\"1\">\r\n<div id=\"fs-id1165042633982\" data-type=\"exercise\"><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Glossary<\/span><\/div>\r\n<\/section>\r\n<div data-type=\"glossary\">\r\n<dl id=\"fs-id1165042832379\">\r\n \t<dt>power reduction formula<\/dt>\r\n \t<dd id=\"fs-id1165042832383\">a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042832387\">\r\n \t<dt>trigonometric integral<\/dt>\r\n \t<dd id=\"fs-id1165042832392\">an integral involving powers and products of trigonometric functions<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1165043207351\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1165043207359\" data-bullet-style=\"bullet\">\n<li>Integrals of trigonometric functions can be evaluated by the use of various strategies. These strategies include:\n<ol id=\"fs-id1165043207368\" type=\"1\">\n<li>Applying trigonometric identities to rewrite the integral so that it may be evaluated by <em data-effect=\"italics\">u<\/em>-substitution<\/li>\n<li>Using integration by parts<\/li>\n<li>Applying trigonometric identities to rewrite products of sines and cosines with different arguments as the sum of individual sine and cosine functions<\/li>\n<li>Applying reduction formulas<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1165043207398\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<p id=\"fs-id1165039565319\">To integrate products involving [latex]\\sin\\left(ax\\right)[\/latex], [latex]\\sin\\left(bx\\right)[\/latex], [latex]\\cos\\left(ax\\right)[\/latex], and [latex]\\cos\\left(bx\\right)[\/latex], use the substitutions.<\/p>\n<ul id=\"fs-id1165042558473\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Sine Products<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\sin\\left(ax\\right)\\sin\\left(bx\\right)=\\frac{1}{2}\\cos\\left(\\left(a-b\\right)x\\right)-\\frac{1}{2}\\cos\\left(\\left(a+b\\right)x\\right)[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Sine and Cosine Products<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\sin\\left(ax\\right)\\cos\\left(bx\\right)=\\frac{1}{2}\\sin\\left(\\left(a-b\\right)x\\right)+\\frac{1}{2}\\sin\\left(\\left(a+b\\right)x\\right)[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Cosine Products<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\cos\\left(ax\\right)\\cos\\left(bx\\right)=\\frac{1}{2}\\cos\\left(\\left(a-b\\right)x\\right)+\\frac{1}{2}\\cos\\left(\\left(a+b\\right)x\\right)[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Power Reduction Formula<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]{\\displaystyle\\int}{\\text{sec}}^{n}xdx=\\frac{1}{n - 1}{\\text{sec}}^{n - 1}x+\\frac{n - 2}{n - 1}{\\displaystyle\\int}{\\text{sec}}^{n - 2}xdx[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Power Reduction Formula<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]{\\displaystyle\\int}{\\tan}^{n}xdx=\\frac{1}{n - 1}{\\tan}^{n - 1}x-{\\displaystyle\\int}{\\tan}^{n - 2}xdx[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1165042637870\" class=\"section-exercises\" data-depth=\"1\">\n<div id=\"fs-id1165042633982\" data-type=\"exercise\"><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Glossary<\/span><\/div>\n<\/section>\n<div data-type=\"glossary\">\n<dl id=\"fs-id1165042832379\">\n<dt>power reduction formula<\/dt>\n<dd id=\"fs-id1165042832383\">a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042832387\">\n<dt>trigonometric integral<\/dt>\n<dd id=\"fs-id1165042832392\">an integral involving powers and products of trigonometric functions<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-668\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 2. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":416434,"menu_order":8,"template":"","meta":{"_candela_citation":"{\"1\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"}}","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-668","chapter","type-chapter","status-publish","hentry"],"part":158,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/668","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/416434"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/668\/revisions"}],"predecessor-version":[{"id":1264,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/668\/revisions\/1264"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/158"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/668\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=668"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=668"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=668"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=668"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}