{"id":190511,"date":"2025-02-12T10:51:42","date_gmt":"2025-02-12T10:51:42","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=190511"},"modified":"2025-02-12T10:51:44","modified_gmt":"2025-02-12T10:51:44","slug":"evaluate-xedx-using-the-integration-by-parts-formula-with-u-x-and-dv-e2dx","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/12\/evaluate-xedx-using-the-integration-by-parts-formula-with-u-x-and-dv-e2dx\/","title":{"rendered":"Evaluate \/ xe*dx using the integration-by-parts formula with u= x and dv = e2dx"},"content":{"rendered":"\n

Evaluate \/ xe*dx using the integration-by-parts formula with u= x and dv = e2<\/em>dx. 3.2 Evaluate \/ xinx dx. 3.3 Evaluate \/ sinxdx.<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

Let’s evaluate each of these integrals step-by-step, using appropriate methods like integration by parts or standard integration rules. I’ll walk you through the solution for each problem and explain the reasoning behind them.<\/p>\n\n\n\n

3.1 Evaluate \u222b x * e^(2x) dx using Integration by Parts<\/h3>\n\n\n\n

We can solve this integral using the integration by parts<\/strong> formula:<\/p>\n\n\n\n

[
\\int u \\, dv = uv – \\int v \\, du
]<\/p>\n\n\n\n

Here, we can choose:<\/p>\n\n\n\n

    \n
  • ( u = x ), so that ( du = dx )<\/li>\n\n\n\n
  • ( dv = e^{2x} dx ), so we need to integrate ( e^{2x} ), which gives ( v = \\frac{1}{2} e^{2x} )<\/li>\n<\/ul>\n\n\n\n

    Now, applying the integration by parts formula:<\/p>\n\n\n\n

    [
    \\int x e^{2x} dx = \\left( x \\cdot \\frac{1}{2} e^{2x} \\right) – \\int \\frac{1}{2} e^{2x} dx
    ]<\/p>\n\n\n\n

    Simplifying:<\/p>\n\n\n\n

    [
    = \\frac{1}{2} x e^{2x} – \\frac{1}{2} \\int e^{2x} dx
    ]<\/p>\n\n\n\n

    The integral of ( e^{2x} ) is ( \\frac{1}{2} e^{2x} ), so:<\/p>\n\n\n\n

    [
    = \\frac{1}{2} x e^{2x} – \\frac{1}{4} e^{2x} + C
    ]<\/p>\n\n\n\n

    Thus, the solution to the integral is:<\/p>\n\n\n\n

    [
    \\int x e^{2x} dx = \\frac{1}{2} x e^{2x} – \\frac{1}{4} e^{2x} + C
    ]<\/p>\n\n\n\n

    3.2 Evaluate \u222b x * sin(x) dx<\/h3>\n\n\n\n

    For this integral, we will again use integration by parts<\/strong>. Let:<\/p>\n\n\n\n

      \n
    • ( u = x ), so ( du = dx )<\/li>\n\n\n\n
    • ( dv = \\sin(x) dx ), so integrating gives ( v = -\\cos(x) )<\/li>\n<\/ul>\n\n\n\n

      Applying the integration by parts formula:<\/p>\n\n\n\n

      [
      \\int x \\sin(x) dx = \\left( x \\cdot -\\cos(x) \\right) – \\int -\\cos(x) dx
      ]<\/p>\n\n\n\n

      Simplifying:<\/p>\n\n\n\n

      [
      = -x \\cos(x) + \\int \\cos(x) dx
      ]<\/p>\n\n\n\n

      The integral of ( \\cos(x) ) is ( \\sin(x) ), so:<\/p>\n\n\n\n

      [
      = -x \\cos(x) + \\sin(x) + C
      ]<\/p>\n\n\n\n

      Thus, the solution to the integral is:<\/p>\n\n\n\n

      [
      \\int x \\sin(x) dx = -x \\cos(x) + \\sin(x) + C
      ]<\/p>\n\n\n\n

      3.3 Evaluate \u222b sin(x) dx<\/h3>\n\n\n\n

      This is a simpler integral that doesn’t require integration by parts. The integral of ( \\sin(x) ) is a standard result:<\/p>\n\n\n\n

      [
      \\int \\sin(x) dx = -\\cos(x) + C
      ]<\/p>\n\n\n\n

      Thus, the solution to the integral is:<\/p>\n\n\n\n

      [
      \\int \\sin(x) dx = -\\cos(x) + C
      ]<\/p>\n\n\n\n

      Explanation<\/h3>\n\n\n\n
        \n
      • Integration by Parts<\/strong>: This technique is useful when you have a product of two functions. The formula ( \\int u \\, dv = uv – \\int v \\, du ) helps break down the product into more manageable parts. By strategically choosing ( u ) and ( dv ), we simplify the integral and solve it step by step.<\/li>\n\n\n\n
      • Basic Integrals<\/strong>: Some integrals, like ( \\int \\sin(x) dx ), are basic and do not require complex methods. These are commonly memorized as standard results.<\/li>\n\n\n\n
      • Practice<\/strong>: Understanding the structure of different integrals\u2014whether they are products of functions or simple trigonometric functions\u2014will help you determine when to apply integration by parts and when a straightforward approach will suffice.<\/li>\n<\/ul>\n\n\n\n

        By following these methods and principles, we can evaluate a wide range of integrals.<\/p>\n","protected":false},"excerpt":{"rendered":"

        Evaluate \/ xe*dx using the integration-by-parts formula with u= x and dv = e2dx. 3.2 Evaluate \/ xinx dx. 3.3 Evaluate \/ sinxdx. The Correct Answer and Explanation is : Let’s evaluate each of these integrals step-by-step, using appropriate methods like integration by parts or standard integration rules. I’ll walk you through the solution for […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-190511","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190511","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=190511"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190511\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=190511"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=190511"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=190511"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}