{"id":400,"date":"2025-02-13T19:44:37","date_gmt":"2025-02-13T19:44:37","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/techniques-for-integration-cheat-sheet\/"},"modified":"2025-02-13T19:44:37","modified_gmt":"2025-02-13T19:44:37","slug":"techniques-for-integration-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/techniques-for-integration-cheat-sheet\/","title":{"raw":"Techniques for Integration: Cheat Sheet","rendered":"Techniques for Integration: Cheat Sheet"},"content":{"raw":"\n<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Techniques+for+Integration.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<p><strong>Integration using Substitution<\/strong><\/p>\n<ul id=\"fs-id1170571098317\">\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>\n\t<li>When using substitution for a definite integral, we also have to change the limits of integration.<\/li>\n<\/ul>\n<p><strong>Integrals Involving Exponential and Logarithmic Functions<\/strong><\/p>\n<ul id=\"fs-id1170571712573\">\n\t<li>Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay.<\/li>\n\t<li>Substitution is often used to evaluate integrals involving exponential functions or logarithms.<\/li>\n<\/ul>\n<p><strong>Integrals Resulting in Inverse Trigonometric Functions<\/strong><\/p>\n<ul id=\"fs-id1170571573238\">\n\t<li>Formulas for derivatives of inverse trigonometric functions developed in Derivatives of Exponential and Logarithmic Functions lead directly to integration formulas involving inverse trigonometric functions.<\/li>\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>\n\t<li>Substitution is often required to put the integrand in the correct form.<\/li>\n<\/ul>\n<p><strong>Approximating Integrals<\/strong><\/p>\n<ul>\n\t<li>All differentiable functions can have derivatives found using established calculus rules.<\/li>\n\t<li>Not all functions can be integrated into a simple antiderivative form using elementary functions.<\/li>\n\t<li>For functions that do not have a straightforward antiderivative, integration can be approximated using methods such as Riemann sums.<\/li>\n\t<li>Riemann sums approximate the area under a curve by dividing the area into rectangles and summing their areas.<\/li>\n\t<li>When approximating integrals, providing upper and lower bounds helps determine the range within which the true value lies.<\/li>\n\t<li>Upper sums use the maximum function value on each subinterval, while lower sums use the minimum function value.<\/li>\n\t<li>The interval between the upper and lower sums gives an estimate of the possible error in the approximation.<\/li>\n\t<li>Larger numbers of subintervals ([latex]n[\/latex]) lead to more accurate approximations.<\/li>\n<\/ul>\n<div id=\"fs-id1170573581296\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170573390237\">\n\t<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\t<li><strong>Substitution with Definite Integrals<\/strong><br>\n[latex]{\\displaystyle\\int }_{a}^{b}f(g(x)){g}^{\\prime }(x)dx={\\displaystyle\\int }_{g(a)}^{g(b)}f(u)du[\/latex]<\/li>\n\t<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\t<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\t<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<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<h2>Study Tips<\/h2>\n<p><strong>Substitution for Indefinite Integrals<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Practice recognizing integrands in the form [latex]f(g(x))g'(x)[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that [latex]du = g'(x)dx[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Don't forget to substitute back to [latex]x[\/latex] at the end<\/li>\n\t<li class=\"whitespace-normal break-words\">Be prepared to adjust constants or solve for [latex]x[\/latex] in terms of [latex]u[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">If the substitution doesn't work, try a different [latex]u[\/latex]<\/li>\n<\/ul>\n<p><strong>Substitution for Definite Integrals<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Practice changing limits of integration when substituting<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember to adjust for any constant factors when substituting<\/li>\n\t<li class=\"whitespace-normal break-words\">Look for opportunities to simplify the integrand before substituting<\/li>\n\t<li class=\"whitespace-normal break-words\">Check your answer by differentiating the result<\/li>\n<\/ul>\n<p><strong>Integrals of Exponential Functions<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Practice identifying when to use direct formula vs. substitution<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that [latex]e^x [\/latex] is its own derivative, which simplifies many integrals<\/li>\n\t<li class=\"whitespace-normal break-words\">For definite integrals, be comfortable with changing limits of integration<\/li>\n<\/ul>\n<p><strong>Integrals&nbsp;Involving Logarithmic Functions<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Practice identifying when to use direct formula vs. substitution<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember the relationship between exponential and logarithmic functions<\/li>\n\t<li class=\"whitespace-normal break-words\">Practice integrating a variety of logarithmic expressions and reciprocal functions<\/li>\n\t<li class=\"whitespace-normal break-words\">Review properties of logarithms to simplify complex expressions before integration<\/li>\n<\/ul>\n<p><strong>Integrals Resulting in Inverse Trigonometric Functions<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Remember domain restrictions for inverse trig functions<\/li>\n\t<li class=\"whitespace-normal break-words\">Review trigonometric identities<\/li>\n\t<li class=\"whitespace-normal break-words\">Practice both indefinite and definite integrals<\/li>\n<\/ul>\n<p><strong>Approximating Integrals<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Practice identifying functions without elementary antiderivatives<\/li>\n\t<li class=\"whitespace-normal break-words\">Understand when to use left-endpoint vs. right-endpoint sums<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that increasing [latex]n[\/latex] generally improves approximation accuracy<\/li>\n\t<li class=\"whitespace-normal break-words\">Be comfortable with calculating and interpreting error bounds<\/li>\n\t<li class=\"whitespace-normal break-words\">Visualize the approximation process using graphs<\/li>\n<\/ul>\n","rendered":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Techniques+for+Integration.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<p><strong>Integration using Substitution<\/strong><\/p>\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<p><strong>Integrals Involving Exponential and Logarithmic Functions<\/strong><\/p>\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<p><strong>Integrals Resulting in Inverse Trigonometric Functions<\/strong><\/p>\n<ul id=\"fs-id1170571573238\">\n<li>Formulas for derivatives of inverse trigonometric functions developed in Derivatives of Exponential and Logarithmic Functions 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<p><strong>Approximating Integrals<\/strong><\/p>\n<ul>\n<li>All differentiable functions can have derivatives found using established calculus rules.<\/li>\n<li>Not all functions can be integrated into a simple antiderivative form using elementary functions.<\/li>\n<li>For functions that do not have a straightforward antiderivative, integration can be approximated using methods such as Riemann sums.<\/li>\n<li>Riemann sums approximate the area under a curve by dividing the area into rectangles and summing their areas.<\/li>\n<li>When approximating integrals, providing upper and lower bounds helps determine the range within which the true value lies.<\/li>\n<li>Upper sums use the maximum function value on each subinterval, while lower sums use the minimum function value.<\/li>\n<li>The interval between the upper and lower sums gives an estimate of the possible error in the approximation.<\/li>\n<li>Larger numbers of subintervals ([latex]n[\/latex]) lead to more accurate approximations.<\/li>\n<\/ul>\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}^{\\prime }(x)dx={\\displaystyle\\int }_{g(a)}^{g(b)}f(u)du[\/latex]<\/li>\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<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<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<h2>Study Tips<\/h2>\n<p><strong>Substitution for Indefinite Integrals<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice recognizing integrands in the form [latex]f(g(x))g'(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Remember that [latex]du = g'(x)dx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Don&#8217;t forget to substitute back to [latex]x[\/latex] at the end<\/li>\n<li class=\"whitespace-normal break-words\">Be prepared to adjust constants or solve for [latex]x[\/latex] in terms of [latex]u[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If the substitution doesn&#8217;t work, try a different [latex]u[\/latex]<\/li>\n<\/ul>\n<p><strong>Substitution for Definite Integrals<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice changing limits of integration when substituting<\/li>\n<li class=\"whitespace-normal break-words\">Remember to adjust for any constant factors when substituting<\/li>\n<li class=\"whitespace-normal break-words\">Look for opportunities to simplify the integrand before substituting<\/li>\n<li class=\"whitespace-normal break-words\">Check your answer by differentiating the result<\/li>\n<\/ul>\n<p><strong>Integrals of Exponential Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying when to use direct formula vs. substitution<\/li>\n<li class=\"whitespace-normal break-words\">Remember that [latex]e^x[\/latex] is its own derivative, which simplifies many integrals<\/li>\n<li class=\"whitespace-normal break-words\">For definite integrals, be comfortable with changing limits of integration<\/li>\n<\/ul>\n<p><strong>Integrals&nbsp;Involving Logarithmic Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying when to use direct formula vs. substitution<\/li>\n<li class=\"whitespace-normal break-words\">Remember the relationship between exponential and logarithmic functions<\/li>\n<li class=\"whitespace-normal break-words\">Practice integrating a variety of logarithmic expressions and reciprocal functions<\/li>\n<li class=\"whitespace-normal break-words\">Review properties of logarithms to simplify complex expressions before integration<\/li>\n<\/ul>\n<p><strong>Integrals Resulting in Inverse Trigonometric Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Remember domain restrictions for inverse trig functions<\/li>\n<li class=\"whitespace-normal break-words\">Review trigonometric identities<\/li>\n<li class=\"whitespace-normal break-words\">Practice both indefinite and definite integrals<\/li>\n<\/ul>\n<p><strong>Approximating Integrals<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying functions without elementary antiderivatives<\/li>\n<li class=\"whitespace-normal break-words\">Understand when to use left-endpoint vs. right-endpoint sums<\/li>\n<li class=\"whitespace-normal break-words\">Remember that increasing [latex]n[\/latex] generally improves approximation accuracy<\/li>\n<li class=\"whitespace-normal break-words\">Be comfortable with calculating and interpreting error bounds<\/li>\n<li class=\"whitespace-normal break-words\">Visualize the approximation process using graphs<\/li>\n<\/ul>\n","protected":false},"author":6,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":399,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/400"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/400\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/399"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/400\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=400"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=400"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=400"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=400"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}