{"id":5845,"date":"2021-10-03T21:30:10","date_gmt":"2021-10-03T16:00:10","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5845"},"modified":"2021-11-25T23:14:14","modified_gmt":"2021-11-25T17:44:14","slug":"integration-of-cos-inverse-x","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-of-cos-inverse-x\/","title":{"rendered":"Integration of Cos Inverse x"},"content":{"rendered":"

Here you will learn proof of integration of cos inverse x or arccos x and examples based on it.<\/p>\n

Let’s begin –<\/p>\n

Integration of Cos Inverse x<\/h2>\n
\n

The integration of cos inverse x or arccos x is \\(xcos^{-1}x\\) – \\(\\sqrt{1 – x^2}\\) + C<\/strong><\/p>\n

Where C is the integration constant.<\/p>\n

i.e. \\(\\int\\) \\(cos^{-1}x\\) = \\(xcos^{-1}x\\) – \\(\\sqrt{1 – x^2}\\) + C<\/p>\n<\/blockquote>\n

Proof :\u00a0<\/strong><\/p>\n

\n

We have, I = \\(\\int\\) \\(cos^{-1}x\\) dx<\/p>\n

Let \\(cos^{-1}x\\) = t,<\/p>\n

Then, x = cos t<\/p>\n

\\(\\implies\\) dx = d(cos t) = -sin t dt<\/p>\n

\\(\\therefore\\) I = \\(\\int\\) \\(cos^{-1}x\\) dx<\/p>\n

\\(\\implies\\) I = \\(\\int\\) -t sint dt<\/p>\n

By using integration by parts formula<\/a>,<\/p>\n

I = t cos t + \\(\\int\\) 1. (-cos t) dt<\/p>\n

I = t cos t – \\(\\int\\) cost dt<\/p>\n

= t cos t – sin t + C<\/p>\n

= t cos t – \\(\\sqrt{1 – cos^2 t}\\) + C<\/p>\n

Now, Put t = \\(cos^{-1}x\\) here<\/p>\n

\\(\\implies\\) I = x \\(cos^{-1}x\\) – \\(\\sqrt{1 – x^2}\\) + C<\/p>\n

Hence, \\(\\int\\) \\(cos^{-1}x\\) dx = x \\(cos^{-1}x\\) – \\(\\sqrt{1 – x^2}\\) + C<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Evaluate \\(\\int\\) \\(x cos^{-1} x\\) dx<\/p>\n

Solution<\/strong><\/span> : We have,<\/p>\n

I = \\(\\int\\)\u00a0 \\(x cos^{-1} x\\) dx<\/p>\n

By using integration by parts formula<\/a>,<\/p>\n

I = \\(cos^{-1} x\\) \\(x^2\\over 2\\) – \\(\\int\\) \\(-1\\over \\sqrt{1 – x^2}\\) \\(\\times\\) \\(x^2\\over 2\\) dx<\/p>\n

I =\u00a0 \\(x^2\\over 2\\) \\(cos^{-1} x\\) – \\(1\\over 2\\) \\(\\int\\) \\(-x^2\\over \\sqrt{1 – x^2}\\) dx<\/p>\n

= \\(x^2\\over 2\\) \\(cos^{-1} x\\) – \\(1\\over 2\\) \\(\\int\\) \\(1 – x^2 – 1\\over \\sqrt{1 – x^2}\\)\u00a0 dx<\/p>\n

= \\(x^2\\over 2\\) \\(cos^{-1} x\\) – \\(1\\over 2\\) { \\(\\int\\) \\(1 – x^2\\over \\sqrt{1 – x^2}\\) – \\(\\int\\) \\(1\\over \\sqrt{1 -x^2}\\) } dx<\/p>\n

\\(\\implies\\) I = \\(x^2\\over 2\\) \\(cos^{-1} x\\) – \\(1\\over 2\\) { \\(\\int\\) \\(\\sqrt{1 – x^2}\\) – \\(\\int\\) \\(1\\over \\sqrt{1 -x^2}\\) } dx<\/p>\n

By using integration formula<\/a> of \\(\\sqrt{a^2 – x^2}\\),<\/p>\n

\\(\\implies\\) I = \\(x^2\\over 2\\) \\(cos^{-1} x\\) – \\(1\\over 2\\) [{ \\(1\\over 2\\) \\(x\\sqrt{1 – x^2}\\) – \\(1\\over 2\\) \\(sin^{-1} x\\) } – \\(sin ^{-1} x\\) ] + C<\/p>\n

\\(\\implies\\) I = \\(x^2\\over 2\\) \\(cos^{-1} x\\) –\u00a0 \\(1\\over 4\\) \\(x\\sqrt{1 – x^2}\\) + \\(3\\over 4\\) \\(sin^{-1} x\\) + C<\/p>\n

\n

Related Questions<\/h3>\n

What is the Differentiation of cos inverse x ?<\/a><\/p>\n

What is the Integration of cos x ?<\/a><\/p>\n

What is the integration of cos inverse root x ?<\/a><\/p>\n\n\n

\n
Next – Integration of Tan Inverse x<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Integration of Sin Inverse x <\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn proof of integration of cos inverse x or arccos x and examples based on it. Let’s begin – Integration of Cos Inverse x The integration of cos inverse x or arccos x is \\(xcos^{-1}x\\) – \\(\\sqrt{1 – x^2}\\) + C Where C is the integration constant. i.e. \\(\\int\\) \\(cos^{-1}x\\) = \\(xcos^{-1}x\\) …<\/p>\n

Integration of Cos Inverse x<\/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":[415,204,414],"yoast_head":"\nIntegration of Cos Inverse x - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn proof of integration of cos inverse x or arccos x and examples based on it.\" \/>\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-cos-inverse-x\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration of Cos Inverse x - 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