{"id":5833,"date":"2021-10-03T22:10:18","date_gmt":"2021-10-03T16:40:18","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5833"},"modified":"2021-11-25T23:13:35","modified_gmt":"2021-11-25T17:43:35","slug":"integration-of-tan-inverse-x","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-of-tan-inverse-x\/","title":{"rendered":"Integration of Tan Inverse x"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

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

The integration of tan inverse x or arctan x is \\(xtan^{-1}x\\) – \\(1\\over 2\\) \\(log |1 + x^2|\\) + C<\/strong><\/p>\n

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

i.e. \\(\\int\\) \\(tan^{-1}x\\) = \\(xtan^{-1}x\\) – \\(1\\over 2\\) \\(log |1 + x^2|\\) + C<\/p>\n<\/blockquote>\n

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

\n

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

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

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

\\(\\implies\\) dx = d(tan t) = \\(sec^2 t\\) dt<\/p>\n

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

\\(\\implies\\) I = \\(\\int\\) t \\(sec^2 t\\) dt<\/p>\n

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

I = t tan t – \\(\\int\\) 1. (tan t) dt<\/p>\n

I = t tan t + log |cos t| + C<\/p>\n

Since tan t = x \\(\\implies\\) cost = \\(1\\over \\sqrt{1 + tan^2 t}\\) = \\(1\\over \\sqrt{1 + x^2}\\)<\/p>\n

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

\\(\\implies\\) I = x \\(tan^{-1}x\\) + \\(log |{1\\over \\sqrt{ 1+ x^2}}|\\) + C<\/p>\n

Hence, \\(\\int\\) \\(tan^{-1}x\\) dx = \\(xtan^{-1}x\\) – \\(1\\over 2\\) \\(log |1 + x^2|\\) + C<\/p>\n<\/blockquote>\n

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

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

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

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

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

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

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

\\(\\implies\\) I = \\(x^2\\over 2\\) \\(tan^{-1} x\\) – \\(1\\over 2\\) (x\u00a0 – \\(tan^{-1} x\\)) + C<\/p>\n

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

\n

Related Questions<\/h3>\n

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

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

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

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

Here you will learn proof of integration of tan inverse x or arctan x and examples based on it. Let’s begin – Integration of Tan Inverse x The integration of tan inverse x or arctan x is \\(xtan^{-1}x\\) – \\(1\\over 2\\) \\(log |1 + x^2|\\) + C Where C is the integration constant. i.e. \\(\\int\\) …<\/p>\n

Integration of Tan 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":[412,204,411],"yoast_head":"\nIntegration of Tan Inverse x - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn proof of integration of tan inverse x or arctan 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-tan-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 Tan Inverse x - 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