{"id":4066,"date":"2021-08-15T18:25:42","date_gmt":"2021-08-15T18:25:42","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4066"},"modified":"2021-11-26T01:29:17","modified_gmt":"2021-11-25T19:59:17","slug":"integration-of-trigonometric-function","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-of-trigonometric-function\/","title":{"rendered":"Integration of Trigonometric Function"},"content":{"rendered":"

Here you will learn integration of trigonometric function with examples.<\/p>\n

Let’s begin –<\/p>\n

Integration of Trigonometric Function<\/h2>\n
\n

(i)<\/strong>\u00a0 \\(\\int\\) \\(sin^mxcos^nx\\)<\/p>\n

Case-1<\/strong> : When both m & n \\(\\in\\) natural numbers.<\/p>\n

(a)\u00a0 If one of them is odd, then substitute for the term of even power.<\/p>\n

(b)\u00a0 If both are odd, substitute either of them.<\/p>\n

(c)\u00a0 If both are even, use trigonometric identities to convert integrand into cosines of multiple angles.<\/p>\n

Case-2<\/strong> : When m + n is a negative even integer.<\/p>\n

In this case the best substitution is tanx = t.<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Evaluate \\(\\int\\) \\(sin^3xcos^5x\\) dx<\/p>\n

Solution : <\/span>We have \\(\\int\\) \\(sin^3xcos^5x\\) dx,

Put cosx = t; -sinx dx = dt

\n so that I = – \\(\\int\\) \\((1 – t^2).t^5\\) dt

\n   = \\(\\int\\) \\((t^7 – t^5)\\) dt = \\(t^8\\over 8\\) – \\(t^6\\over 6\\) = \\(cos^8x\\over 8\\) – \\(cos^6x\\over 6\\) + C

\n <\/p>\n\n\n

\n

(ii)<\/strong>\u00a0 \\(\\int\\) \\(dx\\over {a + bsin^2x}\\) OR \\(\\int\\) \\(dx\\over {a + bcos^2x}\\) OR \\(\\int\\) \\(dx\\over {asin^2x + bsinxcosx + ccos^2x}\\)<\/p>\n

Divide Numerator and Denominator by \\(cos^2x\\) & put tanx = t.<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Evaluate : \\(\\int\\) \\(dx\\over {(2sinx + 3cosx)}^2\\)<\/p>\n

Solution : <\/span>Divide numerator and denominator by \\(cos^2x\\)

\n \\(\\therefore\\)   I = \\(\\int\\) \\(sec^2x dx\\over {(2sinx + 3cosx)}^2\\)

\n Let 2tanx + 3 = t,       \\(\\therefore\\) 2\\(sec^2x\\) dx = dt

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

\n <\/p>\n\n\n

\n

(iii)<\/strong>\u00a0 \\(\\int\\) \\(dx\\over {a + bsinx}\\) OR \\(\\int\\) \\(dx\\over {a + bcosx}\\) OR \\(\\int\\) \\(dx\\over {a + bsinx + ccosx}\\)<\/p>\n

convert sines and cosines into their respective tangents of half the angles and put tan\\(x\\over 2\\) = t<\/p>\n

In this case sinx = \\(2t\\over {1+t^2}\\), cosx = \\({1-t^2}\\over {1+t^2}\\), x = 2\\(tan^{-1}t\\); dx = \\(2dt\\over {1+t^2}\\)<\/p>\n

(iv)\u00a0<\/strong> \\(\\int\\) \\(acosx + bsinx + c\\over {pcosx + qsinx + r}\\)<\/p>\n

Express Numerator = a(Denominator) + b\\(d\\over dx\\)(Denominator) + c & proceed.<\/p>\n<\/blockquote>\n

Hope you learnt integration of trigonometric function, learn more concepts of integration and practice more questions to get ahead in competition. Good Luck!<\/p>\n\n\n

\n
Next – Integration by Partial fraction<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – What is the Formula for Integration by Parts ?<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn integration of trigonometric function with examples. Let’s begin – Integration of Trigonometric Function (i)\u00a0 \\(\\int\\) \\(sin^mxcos^nx\\) Case-1 : When both m & n \\(\\in\\) natural numbers. (a)\u00a0 If one of them is odd, then substitute for the term of even power. (b)\u00a0 If both are odd, substitute either of them. (c)\u00a0 …<\/p>\n

Integration of Trigonometric Function<\/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":[30],"tags":[204,434,435],"yoast_head":"\nIntegration of Trigonometric Function - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn integration of trigonometric function with examples.\" \/>\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-of-trigonometric-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration of Trigonometric Function - 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