{"id":1144,"date":"2021-06-30T17:01:59","date_gmt":"2021-06-30T17:01:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-integrals-resulting-in-inverse-trigonometric-functions\/"},"modified":"2021-11-17T01:45:20","modified_gmt":"2021-11-17T01:45:20","slug":"summary-of-integrals-resulting-in-inverse-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-integrals-resulting-in-inverse-trigonometric-functions\/","title":{"raw":"Summary of Integrals Resulting in Inverse Trigonometric Functions","rendered":"Summary of Integrals Resulting in Inverse Trigonometric Functions"},"content":{"raw":"<div id=\"fs-id1170572380016\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170571573238\">\r\n \t<li>Formulas for derivatives of inverse trigonometric functions developed in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/derivatives-of-exponential-and-logarithmic-functions\/\">Derivatives of Exponential and Logarithmic Functions<\/a> lead directly to integration formulas involving inverse trigonometric functions.<\/li>\r\n \t<li>Use the formulas listed in the rule on integration formulas resulting in inverse trigonometric functions to match up the correct format and make alterations as necessary to solve the problem.<\/li>\r\n \t<li>Substitution is often required to put the integrand in the correct form.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170571602206\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170571602213\">\r\n \t<li><strong>Integrals That Produce Inverse Trigonometric Functions<\/strong>\r\n[latex]\\displaystyle\\int \\frac{du}{\\sqrt{{a}^{2}-{u}^{2}}}={ \\sin }^{-1}\\left(\\frac{u}{a}\\right)+C[\/latex]\r\n[latex]\\displaystyle\\int \\frac{du}{{a}^{2}+{u}^{2}}=\\frac{1}{a}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}\\left(\\frac{u}{a}\\right)+C[\/latex]\r\n[latex]\\displaystyle\\int \\frac{du}{u\\sqrt{{u}^{2}-{a}^{2}}}=\\frac{1}{a}\\phantom{\\rule{0.05em}{0ex}}{ \\sec }^{-1}\\left(\\frac{u}{a}\\right)+C[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<div id=\"fs-id1170572380016\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170571573238\">\n<li>Formulas for derivatives of inverse trigonometric functions developed in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/derivatives-of-exponential-and-logarithmic-functions\/\">Derivatives of Exponential and Logarithmic Functions<\/a> lead directly to integration formulas involving inverse trigonometric functions.<\/li>\n<li>Use the formulas listed in the rule on integration formulas resulting in inverse trigonometric functions to match up the correct format and make alterations as necessary to solve the problem.<\/li>\n<li>Substitution is often required to put the integrand in the correct form.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170571602206\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170571602213\">\n<li><strong>Integrals That Produce Inverse Trigonometric Functions<\/strong><br \/>\n[latex]\\displaystyle\\int \\frac{du}{\\sqrt{{a}^{2}-{u}^{2}}}={ \\sin }^{-1}\\left(\\frac{u}{a}\\right)+C[\/latex]<br \/>\n[latex]\\displaystyle\\int \\frac{du}{{a}^{2}+{u}^{2}}=\\frac{1}{a}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}\\left(\\frac{u}{a}\\right)+C[\/latex]<br \/>\n[latex]\\displaystyle\\int \\frac{du}{u\\sqrt{{u}^{2}-{a}^{2}}}=\\frac{1}{a}\\phantom{\\rule{0.05em}{0ex}}{ \\sec }^{-1}\\left(\\frac{u}{a}\\right)+C[\/latex]<\/li>\n<\/ul>\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-1144\">\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":31,"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-1144","chapter","type-chapter","status-publish","hentry"],"part":1113,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1144","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\/1144\/revisions"}],"predecessor-version":[{"id":2480,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1144\/revisions\/2480"}],"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\/1144\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1144"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1144"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1144"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1144"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}