{"id":200,"date":"2024-10-31T21:16:13","date_gmt":"2024-11-01T01:16:13","guid":{"rendered":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/?post_type=chapter&p=200"},"modified":"2024-10-31T22:55:10","modified_gmt":"2024-11-01T02:55:10","slug":"unit-9-substitution-rule-of-definite-integral","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/chapter\/unit-9-substitution-rule-of-definite-integral\/","title":{"raw":"Unit 9: Substitution Rule for Definite Integrals","rendered":"Unit 9: Substitution Rule for Definite Integrals"},"content":{"raw":"Now that we\u2019ve covered the substitution method for indefinite integrals in Unit 4<\/a>, let\u2019s see how to apply this technique to definite integrals. When using substitution with definite integrals, we not only change the variable but also update the limits of integration to reflect the new variable. That is, the original limits in terms of [latex]x[\/latex]<\/span> are replaced by the corresponding limits in terms of [latex]u[\/latex]<\/span>:\r\n

[latex]\\int_{b}^{a} f(g(x)) g'(x) dx =\\int_{g(b)}^{g(a)} f(u)du[\/latex]<\/span><\/p>\r\nOnce the integral is evaluated in terms of [latex]u[\/latex]<\/span>, the result can be used to compute the definite integral directly, without needing to revert back to the original variable<\/strong>.\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\/2}^1 (1-2x)^5 dx[\/latex]\u00a0 \u00a0 (similar to Q1 in Unit 4<\/a>)\r\n\r\nQ2. [latex]\\int_0^{\\pi\/7} \\cos(7\\theta-5)\u00a0 \\;d\\theta[\/latex]\u00a0 \u00a0 (similar to Q3 in Unit 4<\/a>)\r\n\r\nQ3. [latex]\\int_{-1}^1 x^2 e^{x^3-1} \\;dx[\/latex]\u00a0 \u00a0 \u00a0(similar to Q4 in Unit 4<\/a>)\r\n\r\nQ4. [latex]\\int_{e^2}^{e^3} \\frac{(\\ln x)^6}{x} \\;dx[\/latex]\r\n\r\nQ5. [latex]\\int_{0}^{\\pi} \\sin x \\cos(\\sin x)\u00a0 \\;dx[\/latex]\r\n\r\nQ6. [latex]\\int_{\\pi\/4}^{\\pi\/2} \\cot\\theta \\csc^2\\theta \\;d\\theta[\/latex]\r\n\r\n ","rendered":"

Now that we\u2019ve covered the substitution method for indefinite integrals in Unit 4<\/a>, let\u2019s see how to apply this technique to definite integrals. When using substitution with definite integrals, we not only change the variable but also update the limits of integration to reflect the new variable. That is, the original limits in terms of [latex]x[\/latex]<\/span> are replaced by the corresponding limits in terms of [latex]u[\/latex]<\/span>:<\/p>\n

[latex]\\int_{b}^{a} f(g(x)) g'(x) dx =\\int_{g(b)}^{g(a)} f(u)du[\/latex]<\/span><\/p>\n

Once the integral is evaluated in terms of [latex]u[\/latex]<\/span>, the result can be used to compute the definite integral directly, without needing to revert back to the original variable<\/strong>.<\/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\/2}^1 (1-2x)^5 dx[\/latex]\u00a0 \u00a0 (similar to Q1 in Unit 4<\/a>)<\/p>\n

Q2. [latex]\\int_0^{\\pi\/7} \\cos(7\\theta-5)\u00a0 \\;d\\theta[\/latex]\u00a0 \u00a0 (similar to Q3 in Unit 4<\/a>)<\/p>\n

Q3. [latex]\\int_{-1}^1 x^2 e^{x^3-1} \\;dx[\/latex]\u00a0 \u00a0 \u00a0(similar to Q4 in Unit 4<\/a>)<\/p>\n

Q4. [latex]\\int_{e^2}^{e^3} \\frac{(\\ln x)^6}{x} \\;dx[\/latex]<\/p>\n

Q5. [latex]\\int_{0}^{\\pi} \\sin x \\cos(\\sin x)\u00a0 \\;dx[\/latex]<\/p>\n

Q6. [latex]\\int_{\\pi\/4}^{\\pi\/2} \\cot\\theta \\csc^2\\theta \\;d\\theta[\/latex]<\/p>\n

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