{"id":1987,"date":"2021-08-19T16:07:00","date_gmt":"2021-08-19T16:07:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-other-strategies-for-integration\/"},"modified":"2021-11-19T03:07:28","modified_gmt":"2021-11-19T03:07:28","slug":"skills-review-for-other-strategies-for-integration","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-other-strategies-for-integration\/","title":{"raw":"Skills Review for Other Strategies for Integration","rendered":"Skills Review for Other Strategies for Integration"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcome<\/h3>\r\n<ul>\r\n \t<li>Use substitution to evaluate indefinite integrals<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Other Strategies for Integration section, we will essentially use a table of integration formulas to evaluate integrals. The most important step when using an integration table is to build the exact integration table formula using the integral we are asked to evaluate. This can sometimes be tricky. Here we will review u-substitution techniques that can be useful when building the desired integration table formula.\r\n<h2>Use Substitution to Evaluate Indefinite Integrals<\/h2>\r\n<p id=\"fs-id1170573431694\"><strong><em>Substitution<\/em><\/strong> is where we substitute part of the integrand with the variable [latex]u[\/latex] and part of the integrand with <em>du<\/em>. It is also referred to as <em><strong>change of variables<\/strong><\/em> because we are changing variables to obtain an expression that is easier to work with for applying the integration rules.<\/p>\r\n\r\n<div id=\"fs-id1170573406868\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Substitution with Indefinite Integrals<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170571193831\">Let [latex]u=g(x),,[\/latex] where [latex]{g}^{\\prime }(x)[\/latex] is continuous over an interval, let [latex]f(x)[\/latex] be continuous over the corresponding range of [latex]g[\/latex], and let [latex]F(x)[\/latex] be an antiderivative of [latex]f(x).[\/latex] Then,<\/p>\r\n\r\n<div id=\"fs-id1170573306241\" class=\"equation\" style=\"text-align: center;\">[latex]\\begin{array}{cc} {\\displaystyle\\int f\\left[g(x)\\right]{g}^{\\prime }(x)dx}\\hfill &amp; = {\\displaystyle\\int f(u)du}\\hfill \\\\ &amp; =F(u)+C\\hfill \\\\ &amp; =F(g(x))+C.\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div id=\"fs-id1170573413133\" class=\"bc-section section\">\r\n<p id=\"fs-id1170573363034\">The following steps should be followed when integrating by substitution:<\/p>\r\n\r\n<ol id=\"fs-id1170570997711\">\r\n \t<li>Look carefully at the integrand and select an expression [latex]g(x)[\/latex] within the integrand to set equal to [latex]u[\/latex]. Let\u2019s select [latex]g(x).[\/latex] such that [latex]{g}^{\\prime }(x)[\/latex] is also part of the integrand.<\/li>\r\n \t<li>Substitute [latex]u=g(x)[\/latex] and [latex]du={g}^{\\prime }(x)dx.[\/latex] into the integral.<\/li>\r\n \t<li>We should now be able to evaluate the integral with respect to [latex]u[\/latex]. If the integral can\u2019t be evaluated we need to go back and select a different expression to use as [latex]u[\/latex].<\/li>\r\n \t<li>Evaluate the integral in terms of [latex]u[\/latex].<\/li>\r\n \t<li>Write the result in terms of [latex]x[\/latex] and the expression [latex]g(x).[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating an Indefinite Integral Using Substitution<\/h3>\r\nUse substitution to find the antiderivative of [latex]\\displaystyle\\int 6x{(3{x}^{2}+4)}^{4}dx.[\/latex]\r\n[reveal-answer q=\"fs-id1170572587719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572587719\"]\r\n<p id=\"fs-id1170573569711\">The first step is to choose an expression for [latex]u[\/latex]. We choose [latex]u=3{x}^{2}+4.[\/latex] because then [latex]du=6xdx.,[\/latex] and we already have <em>du<\/em> in the integrand. Write the integral in terms of [latex]u[\/latex]:<\/p>\r\n\r\n<div id=\"fs-id1170570995689\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int 6x{(3{x}^{2}+4)}^{4}dx=\\displaystyle\\int {u}^{4}du.[\/latex]<\/div>\r\n<p id=\"fs-id1170573398341\">Remember that <em>du<\/em> is the derivative of the expression chosen for [latex]u[\/latex], regardless of what is inside the integrand. Now we can evaluate the integral with respect to [latex]u[\/latex]:<\/p>\r\n\r\n<div id=\"fs-id1170571000121\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll} {\\displaystyle\\int {u}^{4}du}\\hfill &amp; =\\frac{{u}^{5}}{5}+C\\hfill \\\\ \\\\ \\\\ &amp; =\\frac{{(3{x}^{2}+4)}^{5}}{5}+C.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170573208347\"><strong>Analysis<\/strong><\/p>\r\n<p id=\"fs-id1170573569153\">We can check our answer by taking the derivative of the result of integration. We should obtain the integrand. Picking a value for <em>C<\/em> of 1, we let [latex]y=\\frac{1}{5}{(3{x}^{2}+4)}^{5}+1.[\/latex] We have<\/p>\r\n\r\n<div id=\"fs-id1170573336352\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=\\frac{1}{5}{(3{x}^{2}+4)}^{5}+1,[\/latex]<\/div>\r\n<p id=\"fs-id1170573411738\">so<\/p>\r\n\r\n<div id=\"fs-id1170573408431\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\hfill {y}^{\\prime }&amp; =(\\frac{1}{5})5{(3{x}^{2}+4)}^{4}6x\\hfill \\\\ &amp; =6x{(3{x}^{2}+4)}^{4}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170573371003\">This is exactly the expression we started with inside the integrand.<\/p>\r\n<span style=\"font-size: 0.9em;\">[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating an Indefinite Integral Using Substitution<\/h3>\r\nUse substitution to find the antiderivative of [latex]\\displaystyle\\int z\\sqrt{{z}^{2}-5}dz.[\/latex]\r\n\r\n[reveal-answer q=\"fs-id1170572587720\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572587720\"]\r\n<div id=\"qfs-id1170573391210\" class=\"hidden-answer\">\r\n<p id=\"fs-id1170573391210\">Rewrite the integral as [latex]\\displaystyle\\int z{({z}^{2}-5)}^{1\\text{\/}2}dz.[\/latex] Let [latex]u={z}^{2}-5[\/latex] and [latex]du=2zdz.[\/latex] Now we have a problem because [latex]du=2zdz[\/latex] and the original expression has only [latex]zdz.[\/latex] We have to alter our expression for <em>du<\/em> or the integral in [latex]u[\/latex] will be twice as large as it should be. If we multiply both sides of the <em>du<\/em> equation by [latex]\\frac{1}{2}.[\/latex] we can solve this problem. Thus,<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\hfill u&amp; ={z}^{2}-5\\hfill \\\\ \\hfill du&amp; =2zdz\\hfill \\\\ \\hfill \\frac{1}{2}du&amp; =\\frac{1}{2}(2z)dz=zdz.\\hfill \\end{array}[\/latex]<\/p>\r\n<p id=\"fs-id1170570976225\">Write the integral in terms of [latex]u[\/latex], but pull the [latex]\\frac{1}{2}[\/latex] outside the integration symbol:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\int z{({z}^{2}-5)}^{1\\text{\/}2}dz=\\frac{1}{2}\\displaystyle\\int {u}^{1\\text{\/}2}du.[\/latex]<\/p>\r\n<p id=\"fs-id1170573586341\">Integrate the expression in [latex]u[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\frac{1}{2} {\\displaystyle\\int {u}^{1\\text{\/}2}du}\\hfill &amp; =(\\frac{1}{2})\\frac{{u}^{3\\text{\/}2}}{\\frac{3}{2}}+C\\hfill \\\\ \\\\ &amp; =(\\frac{1}{2})(\\frac{2}{3}){u}^{3\\text{\/}2}+C\\hfill \\\\ &amp; =\\frac{1}{3}{u}^{3\\text{\/}2}+C\\hfill \\\\ &amp; =\\frac{1}{3}{({z}^{2}-5)}^{3\\text{\/}2}+C.\\hfill \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\"><span style=\"font-size: 0.9em;\">[\/hidden-answer]<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse substitution to find the antiderivative of [latex]\\displaystyle\\int {x}^{2}{({x}^{3}+5)}^{9}dx.[\/latex]\r\n\r\n[reveal-answer q=\"9013327\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"9013327\"]\r\n\r\nMultiply the du equation by [latex]\\frac{1}{3}.[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572628420\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572628420\"]\r\n<div id=\"qfs-id1170573361491\" class=\"hidden-answer\">\r\n<p id=\"fs-id1170573361491\">[latex]\\frac{{({x}^{3}+5)}^{10}}{30}+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcome<\/h3>\n<ul>\n<li>Use substitution to evaluate indefinite integrals<\/li>\n<\/ul>\n<\/div>\n<p>In the Other Strategies for Integration section, we will essentially use a table of integration formulas to evaluate integrals. The most important step when using an integration table is to build the exact integration table formula using the integral we are asked to evaluate. This can sometimes be tricky. Here we will review u-substitution techniques that can be useful when building the desired integration table formula.<\/p>\n<h2>Use Substitution to Evaluate Indefinite Integrals<\/h2>\n<p id=\"fs-id1170573431694\"><strong><em>Substitution<\/em><\/strong> is where we substitute part of the integrand with the variable [latex]u[\/latex] and part of the integrand with <em>du<\/em>. It is also referred to as <em><strong>change of variables<\/strong><\/em> because we are changing variables to obtain an expression that is easier to work with for applying the integration rules.<\/p>\n<div id=\"fs-id1170573406868\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Substitution with Indefinite Integrals<\/h3>\n<hr \/>\n<p id=\"fs-id1170571193831\">Let [latex]u=g(x),,[\/latex] where [latex]{g}^{\\prime }(x)[\/latex] is continuous over an interval, let [latex]f(x)[\/latex] be continuous over the corresponding range of [latex]g[\/latex], and let [latex]F(x)[\/latex] be an antiderivative of [latex]f(x).[\/latex] Then,<\/p>\n<div id=\"fs-id1170573306241\" class=\"equation\" style=\"text-align: center;\">[latex]\\begin{array}{cc} {\\displaystyle\\int f\\left[g(x)\\right]{g}^{\\prime }(x)dx}\\hfill & = {\\displaystyle\\int f(u)du}\\hfill \\\\ & =F(u)+C\\hfill \\\\ & =F(g(x))+C.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1170573413133\" class=\"bc-section section\">\n<p id=\"fs-id1170573363034\">The following steps should be followed when integrating by substitution:<\/p>\n<ol id=\"fs-id1170570997711\">\n<li>Look carefully at the integrand and select an expression [latex]g(x)[\/latex] within the integrand to set equal to [latex]u[\/latex]. Let\u2019s select [latex]g(x).[\/latex] such that [latex]{g}^{\\prime }(x)[\/latex] is also part of the integrand.<\/li>\n<li>Substitute [latex]u=g(x)[\/latex] and [latex]du={g}^{\\prime }(x)dx.[\/latex] into the integral.<\/li>\n<li>We should now be able to evaluate the integral with respect to [latex]u[\/latex]. If the integral can\u2019t be evaluated we need to go back and select a different expression to use as [latex]u[\/latex].<\/li>\n<li>Evaluate the integral in terms of [latex]u[\/latex].<\/li>\n<li>Write the result in terms of [latex]x[\/latex] and the expression [latex]g(x).[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating an Indefinite Integral Using Substitution<\/h3>\n<p>Use substitution to find the antiderivative of [latex]\\displaystyle\\int 6x{(3{x}^{2}+4)}^{4}dx.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572587719\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572587719\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573569711\">The first step is to choose an expression for [latex]u[\/latex]. We choose [latex]u=3{x}^{2}+4.[\/latex] because then [latex]du=6xdx.,[\/latex] and we already have <em>du<\/em> in the integrand. Write the integral in terms of [latex]u[\/latex]:<\/p>\n<div id=\"fs-id1170570995689\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int 6x{(3{x}^{2}+4)}^{4}dx=\\displaystyle\\int {u}^{4}du.[\/latex]<\/div>\n<p id=\"fs-id1170573398341\">Remember that <em>du<\/em> is the derivative of the expression chosen for [latex]u[\/latex], regardless of what is inside the integrand. Now we can evaluate the integral with respect to [latex]u[\/latex]:<\/p>\n<div id=\"fs-id1170571000121\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll} {\\displaystyle\\int {u}^{4}du}\\hfill & =\\frac{{u}^{5}}{5}+C\\hfill \\\\ \\\\ \\\\ & =\\frac{{(3{x}^{2}+4)}^{5}}{5}+C.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170573208347\"><strong>Analysis<\/strong><\/p>\n<p id=\"fs-id1170573569153\">We can check our answer by taking the derivative of the result of integration. We should obtain the integrand. Picking a value for <em>C<\/em> of 1, we let [latex]y=\\frac{1}{5}{(3{x}^{2}+4)}^{5}+1.[\/latex] We have<\/p>\n<div id=\"fs-id1170573336352\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=\\frac{1}{5}{(3{x}^{2}+4)}^{5}+1,[\/latex]<\/div>\n<p id=\"fs-id1170573411738\">so<\/p>\n<div id=\"fs-id1170573408431\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\hfill {y}^{\\prime }& =(\\frac{1}{5})5{(3{x}^{2}+4)}^{4}6x\\hfill \\\\ & =6x{(3{x}^{2}+4)}^{4}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170573371003\">This is exactly the expression we started with inside the integrand.<\/p>\n<p><span style=\"font-size: 0.9em;\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating an Indefinite Integral Using Substitution<\/h3>\n<p>Use substitution to find the antiderivative of [latex]\\displaystyle\\int z\\sqrt{{z}^{2}-5}dz.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572587720\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572587720\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"qfs-id1170573391210\" class=\"hidden-answer\">\n<p id=\"fs-id1170573391210\">Rewrite the integral as [latex]\\displaystyle\\int z{({z}^{2}-5)}^{1\\text{\/}2}dz.[\/latex] Let [latex]u={z}^{2}-5[\/latex] and [latex]du=2zdz.[\/latex] Now we have a problem because [latex]du=2zdz[\/latex] and the original expression has only [latex]zdz.[\/latex] We have to alter our expression for <em>du<\/em> or the integral in [latex]u[\/latex] will be twice as large as it should be. If we multiply both sides of the <em>du<\/em> equation by [latex]\\frac{1}{2}.[\/latex] we can solve this problem. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\hfill u& ={z}^{2}-5\\hfill \\\\ \\hfill du& =2zdz\\hfill \\\\ \\hfill \\frac{1}{2}du& =\\frac{1}{2}(2z)dz=zdz.\\hfill \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1170570976225\">Write the integral in terms of [latex]u[\/latex], but pull the [latex]\\frac{1}{2}[\/latex] outside the integration symbol:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\int z{({z}^{2}-5)}^{1\\text{\/}2}dz=\\frac{1}{2}\\displaystyle\\int {u}^{1\\text{\/}2}du.[\/latex]<\/p>\n<p id=\"fs-id1170573586341\">Integrate the expression in [latex]u[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\frac{1}{2} {\\displaystyle\\int {u}^{1\\text{\/}2}du}\\hfill & =(\\frac{1}{2})\\frac{{u}^{3\\text{\/}2}}{\\frac{3}{2}}+C\\hfill \\\\ \\\\ & =(\\frac{1}{2})(\\frac{2}{3}){u}^{3\\text{\/}2}+C\\hfill \\\\ & =\\frac{1}{3}{u}^{3\\text{\/}2}+C\\hfill \\\\ & =\\frac{1}{3}{({z}^{2}-5)}^{3\\text{\/}2}+C.\\hfill \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\"><span style=\"font-size: 0.9em;\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use substitution to find the antiderivative of [latex]\\displaystyle\\int {x}^{2}{({x}^{3}+5)}^{9}dx.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q9013327\">Hint<\/span><\/p>\n<div id=\"q9013327\" class=\"hidden-answer\" style=\"display: none\">\n<p>Multiply the du equation by [latex]\\frac{1}{3}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572628420\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572628420\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"qfs-id1170573361491\" class=\"hidden-answer\">\n<p id=\"fs-id1170573361491\">[latex]\\frac{{({x}^{3}+5)}^{10}}{30}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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-1987\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/.\">https:\/\/courses.lumenlearning.com\/precalculus\/.<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":349141,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and 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