{"id":9044,"date":"2026-02-21T20:23:32","date_gmt":"2026-02-21T14:53:32","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=9044"},"modified":"2026-02-21T20:23:42","modified_gmt":"2026-02-21T14:53:42","slug":"integration-of-1-1-cosx","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/","title":{"rendered":"integration of 1\/1+cosx , 1\/1 -cos x"},"content":{"rendered":"\n

We can easily find the integration of $\\frac {1}{1+ cos x}$ and $\\frac {1}{1 -cos x}$ using trigonometric identities and fundamental integration techniques, The formula of these integral is given as<\/p>\n\n\n\n

$\\int \\frac {1}{1+ cos x} \\; dx = \\tan \\frac {x}{2} + C$<\/p>\n\n\n\n

or<\/p>\n\n\n\n

$\\int \\frac {1}{1+ cos x} \\; dx = -\\cot x + \\csx x + C$<\/p>\n\n\n\n

$\\int \\frac {1}{1- cos x} \\; dx == -\\cot {x}{2} + C$<\/p>\n\n\n\n

or<\/p>\n\n\n\n

$\\int \\frac {1}{1- cos x} \\; dx = -\\cot x -\\csc x + C$<\/p>\n\n\n\n

Proof of integration of 1\/1+cosx<\/h2>\n\n\n\n

Method I<\/strong><\/p>\n\n\n\n

$\\int \\frac {1}{1+ cos x} \\; dx = \\int \\frac {1}{2\\cos^2 \\frac {x}{2} }$<\/p>\n\n\n\n

$=\\frac {1}{2} \\int \\sec^2 {x}{2} \\; dx$<\/p>\n\n\n\n

Now let ${x}{2} =t$<\/p>\n\n\n\n

Then $dx= 2dt$
Therefore<\/p>\n\n\n\n

$=\\frac {2 }{2} \\int \\sec^2 t \\; dt$ <\/p>\n\n\n\n

$=\\tan t + C$
Substituting back<\/p>\n\n\n\n

$= \\tan {x}{2} + C$<\/p>\n\n\n\n

Method II<\/strong><\/p>\n\n\n\n

$\\int \\frac {1}{1+ cos x} \\; dx = \\int \\frac {1}{1+ cos x} \\times \\frac {1- cos x}{1- cos x}$<\/p>\n\n\n\n

$= \\int \\frac {1- cos x}{1- cos^2 x } \\; dx = \\int \\frac {1-cos x}{\\sin^2 x} \\; dx $<\/p>\n\n\n\n

$=\\int ( \\csc^2 x – \\cot x \\csc x) \\; dx $
$= -\\cot x + \\csx x + C$<\/p>\n\n\n\n

Proof of integration of 1\/1-cosx<\/h2>\n\n\n\n

Method I<\/strong><\/p>\n\n\n\n

$\\int \\frac {1}{1- cos x} \\; dx = \\int \\frac {1}{2\\sin^2 \\frac {x}{2} }$<\/p>\n\n\n\n

$=\\frac {1}{2} \\int \\csc^2 {x}{2} \\; dx$<\/p>\n\n\n\n

Now let ${x}{2} =t$<\/p>\n\n\n\n

Then $dx= 2dt$
Therefore<\/p>\n\n\n\n

$=\\frac {2 }{2} \\int \\csc^2 t \\; dt$ <\/p>\n\n\n\n

$=-\\cot t + C$
Substituting back<\/p>\n\n\n\n

$= -\\cot {x}{2} + C$<\/p>\n\n\n\n

Method II<\/strong><\/p>\n\n\n\n

$\\int \\frac {1}{1- cos x} \\; dx = \\int \\frac {1}{1- cos x} \\times \\frac {1+ cos x}{1+ cos x}$<\/p>\n\n\n\n

$= \\int \\frac {1+ cos x}{1- cos^2 x } \\; dx = \\int \\frac {1+cos x}{\\sin^2 x} \\; dx $<\/p>\n\n\n\n

$=\\int ( \\csc^2 x + \\cot x \\csc x) \\; dx $
$= -\\cot x – \\csx x + C$<\/p>\n\n\n\n

\n

Other Integration Related Articles<\/p>\n\n\n\n

Integration of modulus function<\/a><\/td>Integration of tan x<\/a><\/td><\/td><\/td><\/tr>
Integration of sec x<\/a><\/td>Integration of greatest integer function<\/a><\/td><\/td><\/td><\/tr>
Integration of trigonometric functions<\/a><\/td>Integration of cot x<\/a><\/td><\/td><\/td><\/tr>
Integration of cosec x<\/a><\/td>Integration of sinx<\/a><\/td><\/td><\/td><\/tr>
Integration of irrational functions<\/a><\/td>integration of root x<\/a><\/td><\/td><\/td><\/tr>
Integration of fractional part of x<\/a><\/td>Integration of cos square x<\/a><\/td><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"

We can easily find the integration of $\\frac {1}{1+ cos x}$ and $\\frac {1}{1 -cos x}$ using trigonometric identities and fundamental integration techniques, The formula of these integral is given as $\\int \\frac {1}{1+ cos x} \\; dx = \\tan \\frac {x}{2} + C$ or $\\int \\frac {1}{1+ cos x} \\; dx = -\\cot x […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","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":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-9044","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\nintegration of 1\/1+cosx , 1\/1 -cos x - physicscatalyst's Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"integration of 1\/1+cosx , 1\/1 -cos x - physicscatalyst's Blog\" \/>\n<meta property=\"og:description\" content=\"We can easily find the integration of $frac {1}{1+ cos x}$ and $frac {1}{1 -cos x}$ using trigonometric identities and fundamental integration techniques, The formula of these integral is given as $int frac {1}{1+ cos x} ; dx = tan frac {x}{2} + C$ or $int frac {1}{1+ cos x} ; dx = -cot x […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/\" \/>\n<meta property=\"og:site_name\" content=\"physicscatalyst's Blog\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/PhysicsCatalyst\" \/>\n<meta property=\"article:author\" content=\"https:\/\/www.facebook.com\/PhysicsCatalyst\" \/>\n<meta property=\"article:published_time\" content=\"2026-02-21T14:53:32+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-02-21T14:53:42+00:00\" \/>\n<meta name=\"author\" content=\"physicscatalyst\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"physicscatalyst\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"integration of 1\/1+cosx , 1\/1 -cos x - physicscatalyst's Blog","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:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/","og_locale":"en_US","og_type":"article","og_title":"integration of 1\/1+cosx , 1\/1 -cos x - physicscatalyst's Blog","og_description":"We can easily find the integration of $frac {1}{1+ cos x}$ and $frac {1}{1 -cos x}$ using trigonometric identities and fundamental integration techniques, The formula of these integral is given as $int frac {1}{1+ cos x} ; dx = tan frac {x}{2} + C$ or $int frac {1}{1+ cos x} ; dx = -cot x […]","og_url":"https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/","og_site_name":"physicscatalyst's Blog","article_publisher":"https:\/\/www.facebook.com\/PhysicsCatalyst","article_author":"https:\/\/www.facebook.com\/PhysicsCatalyst","article_published_time":"2026-02-21T14:53:32+00:00","article_modified_time":"2026-02-21T14:53:42+00:00","author":"physicscatalyst","twitter_misc":{"Written by":"physicscatalyst","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/#article","isPartOf":{"@id":"https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/"},"author":{"name":"physicscatalyst","@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/person\/9b302efdc9b32e459cb1e61ab7506d3f"},"headline":"integration of 1\/1+cosx , 1\/1 -cos x","datePublished":"2026-02-21T14:53:32+00:00","dateModified":"2026-02-21T14:53:42+00:00","mainEntityOfPage":{"@id":"https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/"},"wordCount":267,"commentCount":0,"publisher":{"@id":"https:\/\/physicscatalyst.com\/article\/#organization"},"articleSection":["Maths"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/","url":"https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/","name":"integration of 1\/1+cosx , 1\/1 -cos x - physicscatalyst's Blog","isPartOf":{"@id":"https:\/\/physicscatalyst.com\/article\/#website"},"datePublished":"2026-02-21T14:53:32+00:00","dateModified":"2026-02-21T14:53:42+00:00","breadcrumb":{"@id":"https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/physicscatalyst.com\/article\/integration-of-1-1-cosx\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/physicscatalyst.com\/article\/"},{"@type":"ListItem","position":2,"name":"Maths","item":"https:\/\/physicscatalyst.com\/article\/maths\/"},{"@type":"ListItem","position":3,"name":"integration of 1\/1+cosx , 1\/1 -cos x"}]},{"@type":"WebSite","@id":"https:\/\/physicscatalyst.com\/article\/#website","url":"https:\/\/physicscatalyst.com\/article\/","name":"physicscatalyst's Blog","description":"Learn free for class 9th, 10th science\/maths , 12th and IIT-JEE Physics and maths.","publisher":{"@id":"https:\/\/physicscatalyst.com\/article\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/physicscatalyst.com\/article\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/physicscatalyst.com\/article\/#organization","name":"physicscatalyst","url":"https:\/\/physicscatalyst.com\/article\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/logo\/image\/","url":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2024\/08\/cropped-logo-1.jpg","contentUrl":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2024\/08\/cropped-logo-1.jpg","width":96,"height":96,"caption":"physicscatalyst"},"image":{"@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/PhysicsCatalyst","https:\/\/x.com\/physicscatalyst","https:\/\/www.youtube.com\/user\/thephysicscatalyst","https:\/\/www.instagram.com\/physicscatalyst\/"]},{"@type":"Person","@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/person\/9b302efdc9b32e459cb1e61ab7506d3f","name":"physicscatalyst","sameAs":["https:\/\/physicscatalyst.com","https:\/\/www.facebook.com\/PhysicsCatalyst","https:\/\/x.com\/physicscatalyst"]}]}},"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false,"shareaholic-thumbnail":false},"uagb_author_info":{"display_name":"physicscatalyst","author_link":"https:\/\/physicscatalyst.com\/article\/author\/physicscatalyst\/"},"uagb_comment_info":0,"uagb_excerpt":"We can easily find the integration of $\\frac {1}{1+ cos x}$ and $\\frac {1}{1 -cos x}$ using trigonometric identities and fundamental integration techniques, The formula of these integral is given as $\\int \\frac {1}{1+ cos x} \\; dx = \\tan \\frac {x}{2} + C$ or $\\int \\frac {1}{1+ cos x} \\; dx = -\\cot x…","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/9044","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=9044"}],"version-history":[{"count":2,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/9044\/revisions"}],"predecessor-version":[{"id":9046,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/9044\/revisions\/9046"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=9044"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=9044"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=9044"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}