{"id":248520,"date":"2025-07-08T12:46:49","date_gmt":"2025-07-08T12:46:49","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=248520"},"modified":"2025-07-08T12:46:51","modified_gmt":"2025-07-08T12:46:51","slug":"consider-%e2%88%abxsin%e2%81%a1xcos%e2%81%a1xdxfxsin%e2%81%a12x%e2%88%92gxcos%e2%81%a12xc","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/08\/consider-%e2%88%abxsin%e2%81%a1xcos%e2%81%a1xdxfxsin%e2%81%a12x%e2%88%92gxcos%e2%81%a12xc\/","title":{"rendered":"Consider\u00a0\u222bxsin\u2061xcos\u2061xdx=f(x)sin\u20612x\u2212g(x)cos\u20612x+C"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

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

Correct Answer:<\/strong> C<\/p>\n\n\n\n

Explanation:<\/strong><\/p>\n\n\n\n

The problem asks for the value of f(1) + g(1) given the integral equation:
\u222b x sinx cosx dx = f(x) sin2x – g(x) cos2x + C<\/p>\n\n\n\n

To find the functions f(x) and g(x), we first need to solve the integral on the left side.<\/p>\n\n\n\n

    \n
  1. Simplify the Integrand:<\/strong>
    We can simplify the term\u00a0sinx cosx\u00a0using the double-angle identity for sine, which is\u00a0sin(2x) = 2sinx cosx. Rearranging this, we get\u00a0sinx cosx = (1\/2)sin(2x).
    Substituting this into the integral gives:
    \u222b x * (1\/2)sin(2x) dx = (1\/2) \u222b x sin(2x) dx<\/li>\n\n\n\n
  2. Integration by Parts:<\/strong>
    Now, we can solve the integral \u222b x sin(2x) dx using integration by parts. The formula is \u222b u dv = uv – \u222b v du.
    Let’s choose:
    • u = x\u00a0(so\u00a0du = dx)<\/li>
    • dv = sin(2x) dx\u00a0(so\u00a0v = \u222b sin(2x) dx = – (1\/2)cos(2x))<\/li><\/ul>Applying the formula:
      \u222b x sin(2x) dx = x * (-1\/2)cos(2x) – \u222b (-1\/2)cos(2x) dx
      \u222b x sin(2x) dx = – (x\/2)cos(2x) + (1\/2) \u222b cos(2x) dx
      \u222b x sin(2x) dx = – (x\/2)cos(2x) + (1\/2) * (1\/2)sin(2x) + C’
      \u222b x sin(2x) dx = (1\/4)sin(2x) – (x\/2)cos(2x) + C’<\/li>\n\n\n\n
    • Complete the Original Integral:<\/strong>
      Now substitute this result back into our expression from step 1:
      (1\/2) \u222b x sin(2x) dx = (1\/2) [ (1\/4)sin(2x) – (x\/2)cos(2x) + C’ ]
      \u222b x sinx cosx dx = (1\/8)sin(2x) – (x\/4)cos(2x) + C (where C = C’\/2)<\/li>\n\n\n\n
    • Identify f(x) and g(x):<\/strong>
      We are given that \u222b x sinx cosx dx = f(x) sin2x – g(x) cos2x + C.
      Comparing this with our result:
      f(x) sin2x – g(x) cos2x + C = (1\/8)sin(2x) – (x\/4)cos(2x) + CBy matching the coefficients of the trigonometric functions, we find:\n
        \n
      • f(x) = 1\/8<\/li>\n\n\n\n
      • g(x) = x\/4<\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • Calculate f(1) + g(1):<\/strong>
        Now we can evaluate the functions at x = 1:
        • f(1) = 1\/8<\/li>
        • g(1) = 1\/4<\/li><\/ul>Finally, we calculate the sum:
          f(1) + g(1) = 1\/8 + 1\/4 = 1\/8 + 2\/8 = 3\/8<\/li>\n<\/ol>\n\n\n\n

          Therefore, the value of f(1) + g(1) is 3\/8.<\/p>\n\n\n\n

          \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

          The Correct Answer and Explanation is: Correct Answer: C Explanation: The problem asks for the value of f(1) + g(1) given the integral equation:\u222b x sinx cosx dx = f(x) sin2x – g(x) cos2x + C To find the functions f(x) and g(x), we first need to solve the integral on the left side. Therefore, the […]<\/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-248520","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248520","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=248520"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248520\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=248520"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=248520"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=248520"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}