{"id":6002,"date":"2021-10-04T21:48:43","date_gmt":"2021-10-04T16:18:43","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6002"},"modified":"2021-10-04T21:49:29","modified_gmt":"2021-10-04T16:19:29","slug":"integration-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-examples\/","title":{"rendered":"Integration Examples"},"content":{"rendered":"

Here you will learn some integration examples for better understanding of integration concepts<\/a><\/span>.<\/p>\n\n\n

\n
\n\n

Example 1 : <\/span>Evaluate : \\(\\int\\) \\(dx\\over {3sinx + 4cosx}\\)<\/p>\n

Solution : <\/span>I = \\(\\int\\) \\(dx\\over {3sinx + 4cosx}\\) = \\(\\int\\) \\(dx\\over {3[{2tan{x\\over 2}\\over {1+tan^2{x\\over 2}}}] + 4[{1-tan^2{x\\over 2}\\over {1+tan^2{x\\over 2}}}]}\\) = \\(\\int\\) \\(sec^2{x\\over 2}dx\\over {4+6tan{x\\over 2}-4tan^2{x\\over 2}}\\)

\n let \\(tan{x\\over 2}\\) = t,     \\(\\therefore\\)   \\({1\\over 2}sec^2{x\\over 2}\\)dx = dt

\n so I = \\(\\int\\) \\(2dt\\over {4+6t-4t^2}\\) = \\(1\\over 2\\) \\(\\int\\) \\(dt\\over {1-(t^2-{3\\over 2}t})\\) = \\(1\\over 2\\) \\(\\int\\) \\(dt\\over {{25\\over 16}-{(t-{3\\over 4})}^2}\\)

\n = \\(1\\over 2\\) \\(1\\over {2({5\\over 4})}\\) \\(ln|{{{5\\over 4}+(t-{3\\over 4})}\\over {{5\\over 4}-(t-{3\\over 4})}}|\\) + C = \\(1\\over 5\\) \\(ln|{1+2tan{x\\over 2}\\over {4-2tan{x\\over 2}}}|\\) + C<\/p>\n


\n\n

Example 2 : <\/span> Evaluate : \\(\\int\\) \\(cos^4xdx\\over {sin^3x{(sin^5x + cos^5x)^{3\\over 5}}}\\)<\/p>\n

Solution : <\/span>I = \\(\\int\\) \\(cos^4xdx\\over {sin^3x{(sin^5x + cos^5x)^{3\\over 5}}}\\)

\n = \\(\\int\\) \\(cos^4xdx\\over {sin^6x{(1 + cot^5x)^{3\\over 5}}}\\) = \\(\\int\\) \\(cot^4xcosec^2xdx\\over {{(1 + cot^5x)^{3\\over 5}}}\\)

\n Put \\(1+cot^5x\\) = t

\n \\(5cot^4xcosec^2x\\)dx = -dt

\n = -\\(1\\over 5\\) \\(\\int\\) \\(dt\\over {t^{3\/5}}\\) = -\\(1\\over 2\\) \\(t^{2\/5}\\) + C

\n = -\\(1\\over 2\\) \\({(1+cot^5x)}^{2\/5}\\) + C\n <\/p>


\n\n \n

Example 3 : <\/span>Prove that \\(\\int_{0}^{\\pi\/2}\\) log(sinx)dx = \\(\\int_{0}^{\\pi\/2}\\) log(cosx)dx = -\\(\\pi\\over 2\\)log 2.<\/p>\n

Solution : <\/span>Let I = \\(\\int_{0}^{\\pi\/2}\\) log(sinx)dx     …(i)

\n then I = \\(\\int_{0}^{\\pi\/2}\\) \\(log(sin({\\pi\\over 2}-x))\\)dx = \\(\\int_{0}^{\\pi\/2}\\) log(cosx)dx     …(ii)

\n Adding (i) and (ii), we get

\n 2I = \\(\\int_{0}^{\\pi\/2}\\) log(sinx)dx + \\(\\int_{0}^{\\pi\/2}\\) log(cosx)dx = \\(\\int_{0}^{\\pi\/2}\\) (log(sinx)dx + log(cosx))dx

\n \\(\\implies\\)   \\(\\int_{0}^{\\pi\/2}\\) log(sinxcosx)dx = \\(\\int_{0}^{\\pi\/2}\\) \\(log({2sinxcosx\\over 2})\\)dx

\n = \\(\\int_{0}^{\\pi\/2}\\) \\(log({sin2x\\over 2})\\)dx = \\(\\int_{0}^{\\pi\/2}\\) log(sin2x)dx – \\(\\int_{0}^{\\pi\/2}\\) log(2)dx

\n = \\(\\int_{0}^{\\pi\/2}\\) log(sin2x)dx – (log 2)\\({(x)}^{\\pi\/2}_{0}\\)

\n \\(\\implies\\)   2I = \\(\\int_{0}^{\\pi\/2}\\) log(sin2x)dx – \\(\\pi\\over 2\\)log 2     …(iii)

\n Let \\(I_1\\) = \\(\\int_{0}^{\\pi\/2}\\) log(sin2x)dx,   putting 2x = t, we get

\n \\(I_1\\) = \\(\\int_{0}^{\\pi}\\) log(sint)\\(dt\\over 2\\) = \\(1\\over 2\\) \\(\\int_{0}^{\\pi}\\) log(sint)dt = \\(1\\over 2\\) 2\\(\\int_{0}^{\\pi\/2}\\) log(sint)dt

\n \\(I_1\\) = \\(\\int_{0}^{\\pi\/2}\\) log(sinx)dx

\n \\(\\therefore\\)   (iii) becomes; 2I = I – \\(\\pi\\over 2\\)log 2

\n Hence   \\(\\int_{0}^{\\pi\/2}\\) log(sinx)dx = – \\(\\pi\\over 2\\)log 2<\/p>\n
\n

Practice these given integration examples to test your knowledge on concepts of integration.<\/p>\n \n <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn some integration examples for better understanding of integration concepts. Example 1 : Evaluate : \\(\\int\\) \\(dx\\over {3sinx + 4cosx}\\) Solution : I = \\(\\int\\) \\(dx\\over {3sinx + 4cosx}\\) = \\(\\int\\) \\(dx\\over {3[{2tan{x\\over 2}\\over {1+tan^2{x\\over 2}}}] + 4[{1-tan^2{x\\over 2}\\over {1+tan^2{x\\over 2}}}]}\\) = \\(\\int\\) \\(sec^2{x\\over 2}dx\\over {4+6tan{x\\over 2}-4tan^2{x\\over 2}}\\) let \\(tan{x\\over 2}\\) = …<\/p>\n

Integration Examples<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[31,30],"tags":[],"yoast_head":"\nIntegration Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"Solve these Integration examples to practice and test your knowledge of Integration concepts.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/integration-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration Examples - 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