{"id":8548,"date":"2023-12-25T19:39:45","date_gmt":"2023-12-25T14:09:45","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=8548"},"modified":"2024-01-17T23:16:17","modified_gmt":"2024-01-17T17:46:17","slug":"integration-of-cosec-x","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/integration-of-cosec-x\/","title":{"rendered":"Integration of cosec x"},"content":{"rendered":"\n

Integration of cosec x can be found using various integration technique like integration by substitution, Integration by partial fraction along with trigonometric identities. The various formula for integration of cosec x are<\/p>\n\n\n\n

I<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx = \\ln | \\csc(x) – \\cot(x) | + C
\\]<\/p>\n\n\n\n

II<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx = -\\ln | \\csc(x) + \\cot(x) | + C
\\]<\/p>\n\n\n\n

III<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx = \\ln | \\tan \\frac {x}{2} | + C
\\]<\/p>\n\n\n\n

IV<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx = \\frac {1}{2} \\ln | \\frac {\\cos x -1}{\\cos x + 1} | + C
\\]<\/p>\n\n\n\n

Lets check out the proof of each of these<\/p>\n\n\n\n

Proof of I<\/h2>\n\n\n\n

Integration of cosec x can be solved by using a clever trick involving multiplying and dividing by $(\\csc(x) – \\cot(x))$. Here’s how it’s done:<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx = \\int \\csc(x) \\cdot \\frac{\\csc(x) – \\cot(x)}{\\csc(x) -\\cot(x)} \\, dx \\\\
= \\int \\frac{\\csc^2(x) – \\csc(x)\\cot(x)}{\\csc(x) – \\cot(x)} \\, dx.
\\]<\/p>\n\n\n\n

Now, taking u as
$ u= \\csc(x) – \\cot(x) $
$ du =(-\\csc(x)\\cot(x) + \\csc^2(x)) dx $<\/p>\n\n\n\n

Therefore integration becomes<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx =\\int \\frac {1}{u} \\, du,
\\]<\/p>\n\n\n\n

hence<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx = \\ln | \\csc(x) – \\cot(x) | + C
\\]<\/p>\n\n\n\n

where (C) is the constant of integration. This is the integral of $(\\csc(x))$.<\/p>\n\n\n\n

Proof of II<\/h2>\n\n\n\n

We can write the result of I as<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx = \\ln | \\csc(x) – \\cot(x) | + C = \\ln | \\frac {\\csc^2(x) – \\cot^2(x)}{\\csc(x) + \\cot(x) } | + C \\\\
=\\ln | \\frac {1}{\\csc(x) + \\cot(x) } | + C = \\ln | (\\csc(x) + \\cot(x))^{-1} | + C \\\\
=- \\ln | \\csc(x) + \\cot(x) | + C
\\]<\/p>\n\n\n\n

Proof of III<\/h2>\n\n\n\n

\\[
\\int \\csc(x) \\, dx =\\int \\frac {1}{\\sin(x)} \\, dx
=\\int \\frac {1}{2 \\sin(x\/2) \\cos(x\/2)} \\, dx
\\]<\/p>\n\n\n\n

Dividing and Multiplying cos(x\/2) in the denominator<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx = \\frac {1}{2}\\int \\frac {\\sec^2(x\/2)}{\\tan(x\/2)} \\, dx
\\]<\/p>\n\n\n\n

Lets $u=\\tan(x\/2)$
$du= \\frac{1}{2} \\sec^2(x\/2) dx$<\/p>\n\n\n\n

Therefore<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx = \\int \\frac {1}{u} \\, du = ln |u| + C
\\]<\/p>\n\n\n\n

Hence<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx = \\ln | \\tan \\frac {x}{2} | + C
\\]<\/p>\n\n\n\n

Proof of IV<\/h2>\n\n\n\n

This can be proof by using integration by partial fractions<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx =\\int \\frac {1}{\\sin(x)} \\, dx
=\\int \\frac {sin(x)}{\\sin^2(x)} \\, dx
=\\int \\frac {sin(x)}{ 1- \\cos^2(x)} \\, dx
\\]<\/p>\n\n\n\n

Lets $ u=\\cos(x)$
$du= -\\sin(x) dx$<\/p>\n\n\n\n

Therefore<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx =\\int \\frac {1}{u^2 -1} \\, du
=\\frac {1}{2} \\int [\\frac {1}{u-1} – \\frac {1}{u+1} ]\\, du
=\\frac {1}{2} ln |\\frac {u-1}{u+1}| + C
\\]<\/p>\n\n\n\n

Hence<\/p>\n\n\n\n

\\[
\\int \\csc(x) \\, dx = \\frac {1}{2} \\ln | \\frac {\\cos x -1}{\\cos x + 1} | + C
\\]<\/p>\n\n\n\n

Solved Examples<\/h2>\n\n\n\n

Question<\/strong>
Evaluate the definite integral $\\int_{\\pi\/6}^{\\pi\/3} \\csc(x) \\, dx $.
Solution<\/strong><\/p>\n\n\n\n

Using the result from the above<\/p>\n\n\n\n

\\[
\\int_{\\pi\/6}^{\\pi\/3} \\csc(x) \\, dx = \\left[- \\ln | \\csc(x) + \\cot(x) | \\right]_{\\pi\/6}^{\\pi\/3} \\\\
= -\\ln | \\csc(\\pi\/3) + \\cot(\\pi\/3) | + \\ln | \\csc(\\pi\/6) + \\cot(\\pi\/6) | \\\\
= \\ln | 2 |- \\ln | 2 + \\sqrt{3} | \\\\
= \\ln(\\frac {2}{2+ \\sqrt 3}).
\\]<\/p>\n\n\n\n

\n

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

Integration of tan inverse x<\/a><\/td>Integration of cos inverse x<\/a><\/td><\/tr>
Integration of cos square x<\/a><\/td>Integration of fractional part of x<\/a><\/td><\/tr>
integration of uv formula<\/a><\/td>integration of log x<\/a><\/td><\/tr>
integration of sin inverse x<\/a><\/td>Integration of constant term<\/a><\/td><\/tr>
integration of hyperbolic functions<\/a><\/td>List of Integration Formulas <\/a><\/td><\/tr>
integration of log sinx<\/a><\/td>integration of sin x cos x dx<\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"

Integration of cosec x can be found using various integration technique like integration by substitution, Integration by partial fraction along with trigonometric identities. The various formula for integration of cosec x are I \\[\\int \\csc(x) \\, dx = \\ln | \\csc(x) – \\cot(x) | + C\\] II \\[\\int \\csc(x) \\, dx = -\\ln | \\csc(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":[506],"class_list":["post-8548","post","type-post","status-publish","format-standard","hentry","category-maths","tag-integration"],"yoast_head":"\nIntegration of cosec 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-cosec-x\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration of cosec x - physicscatalyst's Blog\" \/>\n<meta property=\"og:description\" content=\"Integration of cosec x can be found using various integration technique like integration by substitution, Integration by partial fraction along with trigonometric identities. 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