{"id":32,"date":"2024-04-16T20:42:25","date_gmt":"2024-04-17T00:42:25","guid":{"rendered":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/?post_type=chapter&p=32"},"modified":"2024-10-31T21:21:45","modified_gmt":"2024-11-01T01:21:45","slug":"substitution-rule-of-indefinite-integrals","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/chapter\/substitution-rule-of-indefinite-integrals\/","title":{"raw":"Unit 4: Substitution Rule of Indefinite Integrals","rendered":"Unit 4: Substitution Rule of Indefinite Integrals"},"content":{"raw":"The Substitution Rule, also known as the u-substitution method, is a pivotal technique in calculus for simplifying the integration of complex functions, especially composite function [latex]f(g(x))[\/latex]<\/span>. It involves substituting a new variable, typically denoted as [latex]u[\/latex], to transform the composite function as\u00a0[latex]f(u)[\/latex], where [latex]u=g(x)[\/latex],<\/span>\u00a0and streamline the integration process.\r\n\r\nBased on the Chain Rule for differentiation, the Substitution Rule leverages the relationship between [latex]u[\/latex]<\/span><\/span><\/span><\/span><\/span> and [latex]x[\/latex]<\/span><\/span><\/span><\/span><\/span> to reconfigure the integral. If [latex]\\mathbf{\\mathcal{u=g(x)}}[\/latex] <\/span>is a differentiable function of [latex]x[\/latex]<\/span>, and [latex]f(u)[\/latex]<\/span>\u00a0is continuous on the range of [latex]u[\/latex]<\/span>, then the rule states that <\/span>\r\n

[latex]\\int f(g(x)) g'(x) dx =\\int f(u)du[\/latex]<\/span><\/span><\/p>\r\nBy rewriting the integral in terms of [latex]u[\/latex]<\/span><\/span><\/span><\/span><\/span>, we can often simplify the expression and make it more manageable.\r\n\r\nThe substitution rule of definite integral can be found at Unit 9<\/a>.\r\n

Practice Questions:<\/strong><\/h6>\r\nEvaluate the following indefinite integrals before referring to the video for comprehensive solutions.\r\n\r\nQ1. [latex]\\int (1-2x)^5 dx[\/latex]\r\n\r\nQ2. [latex]\\int x\\sqrt{x^2-3}\\; dx[\/latex]\r\n\r\nQ3. [latex]\\int \\cos(7\\theta-5)\u00a0 \\;d\\theta[\/latex]\r\n\r\nQ4. [latex]\\int x^2 e^{x^3-1} \\;dx[\/latex]\r\n\r\nQ5. [latex]\\int \\frac{4}{z \\ln^2 z} \\;dz[\/latex]\r\n\r\nQ6. [latex]\\int t \\sqrt{t+2} \\;dt[\/latex]\r\n\r\nQ7. [latex]\\int \\frac{3^{1\/y}}{y^2} \\;dy[\/latex]","rendered":"

The Substitution Rule, also known as the u-substitution method, is a pivotal technique in calculus for simplifying the integration of complex functions, especially composite function [latex]f(g(x))[\/latex]<\/span>. It involves substituting a new variable, typically denoted as [latex]u[\/latex], to transform the composite function as\u00a0[latex]f(u)[\/latex], where [latex]u=g(x)[\/latex],<\/span>\u00a0and streamline the integration process.<\/p>\n

Based on the Chain Rule for differentiation, the Substitution Rule leverages the relationship between [latex]u[\/latex]<\/span><\/span><\/span><\/span><\/span> and [latex]x[\/latex]<\/span><\/span><\/span><\/span><\/span> to reconfigure the integral. If [latex]\\mathbf{\\mathcal{u=g(x)}}[\/latex] <\/span>is a differentiable function of [latex]x[\/latex]<\/span>, and [latex]f(u)[\/latex]<\/span>\u00a0is continuous on the range of [latex]u[\/latex]<\/span>, then the rule states that <\/span><\/p>\n

[latex]\\int f(g(x)) g'(x) dx =\\int f(u)du[\/latex]<\/span><\/span><\/p>\n

By rewriting the integral in terms of [latex]u[\/latex]<\/span><\/span><\/span><\/span><\/span>, we can often simplify the expression and make it more manageable.<\/p>\n

The substitution rule of definite integral can be found at Unit 9<\/a>.<\/p>\n

Practice Questions:<\/strong><\/h6>\n

Evaluate the following indefinite integrals before referring to the video for comprehensive solutions.<\/p>\n

Q1. [latex]\\int (1-2x)^5 dx[\/latex]<\/p>\n

Q2. [latex]\\int x\\sqrt{x^2-3}\\; dx[\/latex]<\/p>\n

Q3. [latex]\\int \\cos(7\\theta-5)\u00a0 \\;d\\theta[\/latex]<\/p>\n

Q4. [latex]\\int x^2 e^{x^3-1} \\;dx[\/latex]<\/p>\n

Q5. [latex]\\int \\frac{4}{z \\ln^2 z} \\;dz[\/latex]<\/p>\n

Q6. [latex]\\int t \\sqrt{t+2} \\;dt[\/latex]<\/p>\n

Q7. [latex]\\int \\frac{3^{1\/y}}{y^2} \\;dy[\/latex]<\/p>\n","protected":false},"author":391,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-32","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/32","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/users\/391"}],"version-history":[{"count":18,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/32\/revisions"}],"predecessor-version":[{"id":205,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/32\/revisions\/205"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/32\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/media?parent=32"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapter-type?post=32"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/contributor?post=32"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/license?post=32"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}