{"id":669,"date":"2021-05-10T18:52:31","date_gmt":"2021-05-10T18:52:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=669"},"modified":"2021-11-17T02:25:47","modified_gmt":"2021-11-17T02:25:47","slug":"summary-of-trigonometric-substitution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-trigonometric-substitution\/","title":{"raw":"Summary of Trigonometric Substitution","rendered":"Summary of Trigonometric Substitution"},"content":{"raw":"<section id=\"fs-id1165042276351\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1165041956823\" data-bullet-style=\"bullet\">\r\n \t<li>For integrals involving [latex]\\sqrt{{a}^{2}-{x}^{2}}[\/latex], use the substitution [latex]x=a\\sin\\theta [\/latex] and [latex]dx=a\\cos\\theta d\\theta [\/latex].<\/li>\r\n \t<li>For integrals involving [latex]\\sqrt{{a}^{2}+{x}^{2}}[\/latex], use the substitution [latex]x=a\\tan\\theta [\/latex] and [latex]dx=a{\\sec}^{2}\\theta d\\theta [\/latex].<\/li>\r\n \t<li>For integrals involving [latex]\\sqrt{{x}^{2}-{a}^{2}}[\/latex], substitute [latex]x=a\\sec\\theta [\/latex] and [latex]dx=a\\sec\\theta \\tan\\theta d\\theta [\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Glossary<\/span>\r\n\r\n<\/section>\r\n<div data-type=\"glossary\">\r\n<dl id=\"fs-id1165041956786\">\r\n \t<dt>trigonometric substitution<\/dt>\r\n \t<dd id=\"fs-id1165041956792\">an integration technique that converts an algebraic integral containing expressions of the form [latex]\\sqrt{{a}^{2}-{x}^{2}}[\/latex], [latex]\\sqrt{{a}^{2}+{x}^{2}}[\/latex], or [latex]\\sqrt{{x}^{2}-{a}^{2}}[\/latex] into a trigonometric integral<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1165042276351\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1165041956823\" data-bullet-style=\"bullet\">\n<li>For integrals involving [latex]\\sqrt{{a}^{2}-{x}^{2}}[\/latex], use the substitution [latex]x=a\\sin\\theta[\/latex] and [latex]dx=a\\cos\\theta d\\theta[\/latex].<\/li>\n<li>For integrals involving [latex]\\sqrt{{a}^{2}+{x}^{2}}[\/latex], use the substitution [latex]x=a\\tan\\theta[\/latex] and [latex]dx=a{\\sec}^{2}\\theta d\\theta[\/latex].<\/li>\n<li>For integrals involving [latex]\\sqrt{{x}^{2}-{a}^{2}}[\/latex], substitute [latex]x=a\\sec\\theta[\/latex] and [latex]dx=a\\sec\\theta \\tan\\theta d\\theta[\/latex].<\/li>\n<\/ul>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Glossary<\/span><\/p>\n<\/section>\n<div data-type=\"glossary\">\n<dl id=\"fs-id1165041956786\">\n<dt>trigonometric substitution<\/dt>\n<dd id=\"fs-id1165041956792\">an integration technique that converts an algebraic integral containing expressions of the form [latex]\\sqrt{{a}^{2}-{x}^{2}}[\/latex], [latex]\\sqrt{{a}^{2}+{x}^{2}}[\/latex], or [latex]\\sqrt{{x}^{2}-{a}^{2}}[\/latex] into a trigonometric integral<\/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-669\">\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":11,"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-669","chapter","type-chapter","status-publish","hentry"],"part":158,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/669","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":5,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/669\/revisions"}],"predecessor-version":[{"id":2542,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/669\/revisions\/2542"}],"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\/669\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=669"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=669"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=669"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=669"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}