{"id":1141,"date":"2021-06-30T17:01:59","date_gmt":"2021-06-30T17:01:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-integrals-involving-exponential-and-logarithmic-functions\/"},"modified":"2021-11-17T01:44:30","modified_gmt":"2021-11-17T01:44:30","slug":"summary-of-integrals-involving-exponential-and-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-integrals-involving-exponential-and-logarithmic-functions\/","title":{"raw":"Summary of Integrals Involving Exponential and Logarithmic Functions","rendered":"Summary of Integrals Involving Exponential and Logarithmic Functions"},"content":{"raw":"<div id=\"fs-id1170571712566\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170571712573\">\r\n \t<li>Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay.<\/li>\r\n \t<li>Substitution is often used to evaluate integrals involving exponential functions or logarithms.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170571712590\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>Integrals of Exponential Functions<\/strong>\r\n[latex]\\displaystyle\\int {e}^{x}dx={e}^{x}+C[\/latex]\r\n[latex]\\displaystyle\\int {a}^{x}dx=\\frac{{a}^{x}}{\\text{ln}a}+C[\/latex]<\/li>\r\n \t<li><strong>Integration Formulas Involving Logarithmic Functions<\/strong>\r\n[latex]\\displaystyle\\int {x}^{-1}dx=\\text{ln}|x|+C[\/latex]\r\n[latex]\\displaystyle\\int \\text{ln}xdx=x\\text{ln}x-x+C=x(\\text{ln}x-1)+C[\/latex]\r\n[latex]\\displaystyle\\int {\\text{log}}_{a}xdx=\\frac{x}{\\text{ln}a}(\\text{ln}x-1)+C[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<div id=\"fs-id1170571712566\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170571712573\">\n<li>Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay.<\/li>\n<li>Substitution is often used to evaluate integrals involving exponential functions or logarithms.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170571712590\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul>\n<li><strong>Integrals of Exponential Functions<\/strong><br \/>\n[latex]\\displaystyle\\int {e}^{x}dx={e}^{x}+C[\/latex]<br \/>\n[latex]\\displaystyle\\int {a}^{x}dx=\\frac{{a}^{x}}{\\text{ln}a}+C[\/latex]<\/li>\n<li><strong>Integration Formulas Involving Logarithmic Functions<\/strong><br \/>\n[latex]\\displaystyle\\int {x}^{-1}dx=\\text{ln}|x|+C[\/latex]<br \/>\n[latex]\\displaystyle\\int \\text{ln}xdx=x\\text{ln}x-x+C=x(\\text{ln}x-1)+C[\/latex]<br \/>\n[latex]\\displaystyle\\int {\\text{log}}_{a}xdx=\\frac{x}{\\text{ln}a}(\\text{ln}x-1)+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-1141\">\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":28,"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-1141","chapter","type-chapter","status-publish","hentry"],"part":1113,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1141","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\/1141\/revisions"}],"predecessor-version":[{"id":2478,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1141\/revisions\/2478"}],"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\/1141\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1141"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1141"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1141"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1141"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}