{"id":666,"date":"2021-05-10T18:49:56","date_gmt":"2021-05-10T18:49:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=666"},"modified":"2021-11-17T02:24:08","modified_gmt":"2021-11-17T02:24:08","slug":"summary-of-integration-by-parts","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-integration-by-parts\/","title":{"raw":"Summary of Integration by Parts","rendered":"Summary of Integration by Parts"},"content":{"raw":"<section id=\"fs-id1165042299703\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1165042299710\" data-bullet-style=\"bullet\">\r\n \t<li>The integration-by-parts formula allows the exchange of one integral for another, possibly easier, integral.<\/li>\r\n \t<li>Integration by parts applies to both definite and indefinite integrals.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165042299726\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1165042299733\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Integration by parts formula<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\displaystyle\\int udv=uv-\\displaystyle\\int vdu[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Integration by parts for definite integrals<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]{\\displaystyle\\int }_{a}^{b}udv={uv|}_{a}^{b}-{\\displaystyle\\int }_{a}^{b}vdu[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165040639374\">\r\n \t<dt>integration by parts<\/dt>\r\n \t<dd id=\"fs-id1165040639379\">a technique of integration that allows the exchange of one integral for another using the formula [latex]{\\displaystyle\\int}udv=uv-{\\displaystyle\\int}vdu[\/latex]<\/dd>\r\n<\/dl>","rendered":"<section id=\"fs-id1165042299703\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1165042299710\" data-bullet-style=\"bullet\">\n<li>The integration-by-parts formula allows the exchange of one integral for another, possibly easier, integral.<\/li>\n<li>Integration by parts applies to both definite and indefinite integrals.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1165042299726\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1165042299733\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Integration by parts formula<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\displaystyle\\int udv=uv-\\displaystyle\\int vdu[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Integration by parts for definite integrals<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]{\\displaystyle\\int }_{a}^{b}udv={uv|}_{a}^{b}-{\\displaystyle\\int }_{a}^{b}vdu[\/latex]<\/li>\n<\/ul>\n<\/section>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165040639374\">\n<dt>integration by parts<\/dt>\n<dd id=\"fs-id1165040639379\">a technique of integration that allows the exchange of one integral for another using the formula [latex]{\\displaystyle\\int}udv=uv-{\\displaystyle\\int}vdu[\/latex]<\/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-666\">\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":4,"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-666","chapter","type-chapter","status-publish","hentry"],"part":158,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/666","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":4,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/666\/revisions"}],"predecessor-version":[{"id":2540,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/666\/revisions\/2540"}],"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\/666\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=666"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=666"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=666"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=666"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}