{"id":209,"date":"2024-10-31T21:31:24","date_gmt":"2024-11-01T01:31:24","guid":{"rendered":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/?post_type=chapter&p=209"},"modified":"2024-10-31T22:54:59","modified_gmt":"2024-11-01T02:54:59","slug":"unit-10-integration-by-parts-for-definite-integral","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/chapter\/unit-10-integration-by-parts-for-definite-integral\/","title":{"raw":"Unit 10: Integration by Parts for Definite Integrals","rendered":"Unit 10: Integration by Parts for Definite Integrals"},"content":{"raw":"Having introduced the integration by parts technique for indefinite integrals in Unit 5<\/a>, let\u2019s now extend this method to definite integrals. When applying integration by parts to a definite integral, we use the same formula, but with limits of integration included:\r\n\r\n[latex]\\int_a^b u dv = [uv]_a^b - \\int_a^b v du[\/latex]\r\n\r\nHere\u2019s the procedure:\r\n
    \r\n \t
  1. Choose [latex]u[\/latex] and [latex]dv[\/latex]<\/strong>, differentiating [latex]u[\/latex] to find [latex]du[\/latex] and integrating [latex]dv[\/latex] to find [latex]v[\/latex]. This step is same as in Unit 5<\/a>.<\/li>\r\n \t
  2. Evaluate [latex]uv[\/latex] at the boundaries<\/strong> [latex]a[\/latex] and [latex]b[\/latex], i.e.\u00a0 [latex][uv]_a^b=u(b)v(b)-u(a)v(a)[\/latex]<\/li>\r\n \t
  3. Integrate<\/strong> the remaining integral [latex]\\int_a^b v du[\/latex]<\/strong> within the limits [latex]a[\/latex] and [latex]b[\/latex].<\/li>\r\n<\/ol>\r\n
    \r\n
    \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_0^\\pi \\theta \\cos\\theta\\;d\\theta[\/latex]\u00a0 \u00a0 (similar to Q1 in Unit 5<\/a>)\r\n\r\nQ2. [latex]\\int_1^e \\ln x\\;dx[\/latex]\u00a0 \u00a0(similar to Q2 in Unit 5<\/a>)\r\n\r\n<\/div>\r\n<\/div>\r\nQ3. [latex]\\int_{-1}^{3} x^2e^x\\;dx[\/latex]\u00a0 \u00a0 (similar to Q3 in Unit 5<\/a>)\r\n\r\nQ4. [latex]\\int_{-\\pi\/4}^{\\pi\/2} \\cos^3 x \\;dx[\/latex]\u00a0 \u00a0 (similar to Q5 in Unit 5<\/a>)\r\n\r\nQ5. [latex]\\int_1^7\u00a0 t \\sqrt{t+2} \\;dt[\/latex]\u00a0 \u00a0 (similar to Q6 in Unit 5<\/a>)","rendered":"

    Having introduced the integration by parts technique for indefinite integrals in Unit 5<\/a>, let\u2019s now extend this method to definite integrals. When applying integration by parts to a definite integral, we use the same formula, but with limits of integration included:<\/p>\n

    [latex]\\int_a^b u dv = [uv]_a^b - \\int_a^b v du[\/latex]<\/p>\n

    Here\u2019s the procedure:<\/p>\n

      \n
    1. Choose [latex]u[\/latex] and [latex]dv[\/latex]<\/strong>, differentiating [latex]u[\/latex] to find [latex]du[\/latex] and integrating [latex]dv[\/latex] to find [latex]v[\/latex]. This step is same as in Unit 5<\/a>.<\/li>\n
    2. Evaluate [latex]uv[\/latex] at the boundaries<\/strong> [latex]a[\/latex] and [latex]b[\/latex], i.e.\u00a0 [latex][uv]_a^b=u(b)v(b)-u(a)v(a)[\/latex]<\/li>\n
    3. Integrate<\/strong> the remaining integral [latex]\\int_a^b v du[\/latex]<\/strong> within the limits [latex]a[\/latex] and [latex]b[\/latex].<\/li>\n<\/ol>\n
      \n
      \n
      Practice Questions:<\/strong><\/h6>\n

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

      Q1. [latex]\\int_0^\\pi \\theta \\cos\\theta\\;d\\theta[\/latex]\u00a0 \u00a0 (similar to Q1 in Unit 5<\/a>)<\/p>\n

      Q2. [latex]\\int_1^e \\ln x\\;dx[\/latex]\u00a0 \u00a0(similar to Q2 in Unit 5<\/a>)<\/p>\n<\/div>\n<\/div>\n

      Q3. [latex]\\int_{-1}^{3} x^2e^x\\;dx[\/latex]\u00a0 \u00a0 (similar to Q3 in Unit 5<\/a>)<\/p>\n

      Q4. [latex]\\int_{-\\pi\/4}^{\\pi\/2} \\cos^3 x \\;dx[\/latex]\u00a0 \u00a0 (similar to Q5 in Unit 5<\/a>)<\/p>\n

      Q5. [latex]\\int_1^7\u00a0 t \\sqrt{t+2} \\;dt[\/latex]\u00a0 \u00a0 (similar to Q6 in Unit 5<\/a>)<\/p>\n","protected":false},"author":391,"menu_order":10,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-209","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/209","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":13,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/209\/revisions"}],"predecessor-version":[{"id":237,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/209\/revisions\/237"}],"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\/209\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/media?parent=209"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapter-type?post=209"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/contributor?post=209"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/license?post=209"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}