{"id":5839,"date":"2021-10-03T22:55:21","date_gmt":"2021-10-03T17:25:21","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5839"},"modified":"2021-11-25T23:16:21","modified_gmt":"2021-11-25T17:46:21","slug":"integration-of-sec-inverse-x-and-cosec-inverse-x","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-of-sec-inverse-x-and-cosec-inverse-x\/","title":{"rendered":"Integration of Sec Inverse x and Cosec Inverse x"},"content":{"rendered":"

Here you will learn proof of integration of sec inverse x\u00a0 and cosec inverse x.<\/p>\n

Let’s begin –<\/p>\n

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

The integration of sec inverse x or arcsec x is \\(xsec^{-1}x\\) – \\(log|x + \\sqrt{x^2 – 1}|\\) + C<\/strong><\/p>\n

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

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

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

\n

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

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

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

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

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

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

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

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

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

Since tan t = \\(\\sqrt{sec^2 t – 1}\\)\u00a0<\/p>\n

\\(\\implies\\) I = t sec t – \\(log |sec t + \\sqrt{sec^2t – 1}|\\) + C<\/p>\n

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

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

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

Integration of Cosec Inverse x<\/strong><\/h2>\n
\n

The integration of cosec inverse x or arccosec x is x\\(cosec^{-1}x\\) – \\(log|x – \\sqrt{x^2 – 1}|\\) + C<\/strong><\/p>\n

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

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

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

\n

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

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

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

\\(\\implies\\) dx = d(cosec t) = -cosec t cot t dt<\/p>\n

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

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

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

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

I = t cosec t – log |cosec – cot t| + C<\/p>\n

Since cot t = \\(\\sqrt{cosec^2 t – 1}\\)\u00a0<\/p>\n

\\(\\implies\\) I = t cosec t – \\(log |cosec t – \\sqrt{cosec^2t – 1}|\\) + C<\/p>\n

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

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

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

\n

Related Questions<\/h3>\n

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

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

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

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

Here you will learn proof of integration of sec inverse x\u00a0 and cosec inverse x. Let’s begin – Integration of Sec Inverse x The integration of sec inverse x or arcsec x is \\(xsec^{-1}x\\) – \\(log|x + \\sqrt{x^2 – 1}|\\) + C Where C is the integration constant. i.e. \\(\\int\\) \\(sec^{-1}x\\) = \\(xsec^{-1}x\\) – \\(log|x …<\/p>\n

Integration of Sec Inverse x and Cosec 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":[204,408,407],"yoast_head":"\nIntegration of Sec Inverse x and Cosec Inverse x - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the integration of sec inverse x and integration of cosec inverse x and its proof.\" \/>\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-sec-inverse-x-and-cosec-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 Sec Inverse x and Cosec Inverse x - 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