{"id":5841,"date":"2021-10-03T22:54:52","date_gmt":"2021-10-03T17:24:52","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5841"},"modified":"2021-11-25T23:04:06","modified_gmt":"2021-11-25T17:34:06","slug":"integration-of-cot-inverse-x","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/","title":{"rendered":"Integration of Cot Inverse x"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

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

The integration of cot inverse x or arccot x is \\(xcot^{-1}x\\) + \\(1\\over 2\\) \\(log |1 + x^2|\\) + C<\/strong><\/p>\n

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

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

Proof : <\/strong><\/p>\n

\n

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

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

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

\\(\\implies\\) dx = d(cot t) = \\(-cosec^2 t\\) dt<\/p>\n

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

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

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

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

I = t cot t – log |sin t| + C<\/p>\n

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

Hence, sin t = \\(1\\over \\sqrt{1 + x^2}\\)<\/p>\n

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

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

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

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

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

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

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

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

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

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

\n

Related Questions<\/h3>\n

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

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

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

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

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

Integration of Cot 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":[409,413,204,410],"yoast_head":"\nIntegration of Cot Inverse x - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn proof of integration of cot inverse x or arccot 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-cot-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 Cot Inverse x - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn proof of integration of cot inverse x or arccot x and examples based on it.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-03T17:24:52+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-25T17:34:06+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Integration of Cot Inverse x\",\"datePublished\":\"2021-10-03T17:24:52+00:00\",\"dateModified\":\"2021-11-25T17:34:06+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/\"},\"wordCount\":322,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"cot inverse x formula\",\"integral of cot inverse x\",\"integration\",\"integration of cot inverse x\"],\"articleSection\":[\"Indefinite Integration\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/\",\"url\":\"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/\",\"name\":\"Integration of Cot Inverse x - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-03T17:24:52+00:00\",\"dateModified\":\"2021-11-25T17:34:06+00:00\",\"description\":\"In this post you will learn proof of integration of cot inverse x or arccot x and examples based on it.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Integration of Cot Inverse x\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - Study Math Online\",\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathemerize.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathemerize.com\/#organization\",\"name\":\"Mathemerize\",\"url\":\"https:\/\/mathemerize.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"width\":140,\"height\":96,\"caption\":\"Mathemerize\"},\"image\":{\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.instagram.com\/mathemerize\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\",\"name\":\"mathemerize\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"caption\":\"mathemerize\"},\"sameAs\":[\"https:\/\/mathemerize.com\"],\"url\":\"https:\/\/mathemerize.com\/author\/mathemerize\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Integration of Cot Inverse x - Mathemerize","description":"In this post you will learn proof of integration of cot inverse x or arccot x and examples based on it.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/","og_locale":"en_US","og_type":"article","og_title":"Integration of Cot Inverse x - Mathemerize","og_description":"In this post you will learn proof of integration of cot inverse x or arccot x and examples based on it.","og_url":"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/","og_site_name":"Mathemerize","article_published_time":"2021-10-03T17:24:52+00:00","article_modified_time":"2021-11-25T17:34:06+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"Integration of Cot Inverse x","datePublished":"2021-10-03T17:24:52+00:00","dateModified":"2021-11-25T17:34:06+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/"},"wordCount":322,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"keywords":["cot inverse x formula","integral of cot inverse x","integration","integration of cot inverse x"],"articleSection":["Indefinite Integration"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/","url":"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/","name":"Integration of Cot Inverse x - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-10-03T17:24:52+00:00","dateModified":"2021-11-25T17:34:06+00:00","description":"In this post you will learn proof of integration of cot inverse x or arccot x and examples based on it.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/integration-of-cot-inverse-x\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"Integration of Cot Inverse x"}]},{"@type":"WebSite","@id":"https:\/\/mathemerize.com\/#website","url":"https:\/\/mathemerize.com\/","name":"Mathemerize","description":"Maths Tutorials - Study Math Online","publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathemerize.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/mathemerize.com\/#organization","name":"Mathemerize","url":"https:\/\/mathemerize.com\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/","url":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","contentUrl":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","width":140,"height":96,"caption":"Mathemerize"},"image":{"@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.instagram.com\/mathemerize\/"]},{"@type":"Person","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df","name":"mathemerize","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","caption":"mathemerize"},"sameAs":["https:\/\/mathemerize.com"],"url":"https:\/\/mathemerize.com\/author\/mathemerize\/"}]}},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/5841"}],"collection":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/comments?post=5841"}],"version-history":[{"count":8,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/5841\/revisions"}],"predecessor-version":[{"id":8623,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/5841\/revisions\/8623"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=5841"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=5841"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=5841"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}