{"id":1137,"date":"2021-06-30T17:01:58","date_gmt":"2021-06-30T17:01:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-substitution\/"},"modified":"2021-11-17T01:42:44","modified_gmt":"2021-11-17T01:42:44","slug":"summary-of-substitution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-substitution\/","title":{"raw":"Summary of Substitution","rendered":"Summary of Substitution"},"content":{"raw":"<div id=\"fs-id1170571220845\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170571098317\">\r\n \t<li>Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term \u2018substitution\u2019 refers to changing variables or substituting the variable [latex]u[\/latex] and <em>du<\/em> for appropriate expressions in the integrand.<\/li>\r\n \t<li>When using substitution for a definite integral, we also have to change the limits of integration.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170573581296\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170573390237\">\r\n \t<li><strong>Substitution with Indefinite Integrals<\/strong>\r\n[latex]\\displaystyle\\int f\\left[g(x)\\right]{g}^{\\prime }(x)dx=\\displaystyle\\int f(u)du=F(u)+C=F(g(x))+C[\/latex]<\/li>\r\n \t<li><strong>Substitution with Definite Integrals<\/strong>\r\n[latex]{\\displaystyle\\int }_{a}^{b}f(g(x))g\\text{\u2018}(x)dx={\\displaystyle\\int }_{g(a)}^{g(b)}f(u)du[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170573497269\" class=\"definition\">\r\n \t<dt>change of variables<\/dt>\r\n \t<dd id=\"fs-id1170573497274\">the substitution of a variable, such as [latex]u[\/latex], for an expression in the integrand<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573497284\" class=\"definition\">\r\n \t<dt>integration by substitution<\/dt>\r\n \t<dd id=\"fs-id1170573497289\">a technique for integration that allows integration of functions that are the result of a chain-rule derivative<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1170571220845\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170571098317\">\n<li>Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term \u2018substitution\u2019 refers to changing variables or substituting the variable [latex]u[\/latex] and <em>du<\/em> for appropriate expressions in the integrand.<\/li>\n<li>When using substitution for a definite integral, we also have to change the limits of integration.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170573581296\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170573390237\">\n<li><strong>Substitution with Indefinite Integrals<\/strong><br \/>\n[latex]\\displaystyle\\int f\\left[g(x)\\right]{g}^{\\prime }(x)dx=\\displaystyle\\int f(u)du=F(u)+C=F(g(x))+C[\/latex]<\/li>\n<li><strong>Substitution with Definite Integrals<\/strong><br \/>\n[latex]{\\displaystyle\\int }_{a}^{b}f(g(x))g\\text{\u2018}(x)dx={\\displaystyle\\int }_{g(a)}^{g(b)}f(u)du[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170573497269\" class=\"definition\">\n<dt>change of variables<\/dt>\n<dd id=\"fs-id1170573497274\">the substitution of a variable, such as [latex]u[\/latex], for an expression in the integrand<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573497284\" class=\"definition\">\n<dt>integration by substitution<\/dt>\n<dd id=\"fs-id1170573497289\">a technique for integration that allows integration of functions that are the result of a chain-rule derivative<\/dd>\n<\/dl>\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-1137\">\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":17533,"menu_order":24,"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-1137","chapter","type-chapter","status-publish","hentry"],"part":1113,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1137","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\/17533"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1137\/revisions"}],"predecessor-version":[{"id":2475,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1137\/revisions\/2475"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/1113"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1137\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1137"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1137"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1137"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}