{"id":7784,"date":"2021-11-06T14:04:22","date_gmt":"2021-11-06T08:34:22","guid":{"rendered":"https:\/\/mathemerize.com\/?p=7784"},"modified":"2021-11-06T15:36:17","modified_gmt":"2021-11-06T10:06:17","slug":"what-is-the-integration-of-x-cos-inverse-x","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-the-integration-of-x-cos-inverse-x\/","title":{"rendered":"What is the integration of x cos inverse x ?"},"content":{"rendered":"

Solution :<\/h2>\n

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

By using integration by parts formula,<\/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 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

Similar Questions<\/h3>\n

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

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

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

What is the integration of sin inverse x whole square ?<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : We have, I = \\(\\int\\)\u00a0 \\(x cos^{-1} x\\) dx By using integration by parts formula, I = \\(cos^{-1} x\\) \\(x^2\\over 2\\) – \\(\\int\\) \\(-1\\over \\sqrt{1 – x^2}\\) \\(\\times\\) \\(x^2\\over 2\\) dx I =\u00a0 \\(x^2\\over 2\\) \\(cos^{-1} x\\) – \\(1\\over 2\\) \\(\\int\\) \\(-x^2\\over \\sqrt{1 – x^2}\\) dx = \\(x^2\\over 2\\) \\(cos^{-1} x\\) – \\(1\\over …<\/p>\n

What is the integration of x 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":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[52,43],"tags":[],"yoast_head":"\nWhat is the integration of x cos inverse x ? - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the integration of x cos inverse x by using integration by parts.\" \/>\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\/what-is-the-integration-of-x-cos-inverse-x\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is the integration of x cos inverse x ? 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