{"id":5829,"date":"2021-10-03T18:40:10","date_gmt":"2021-10-03T13:10:10","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5829"},"modified":"2021-11-25T23:52:06","modified_gmt":"2021-11-25T18:22:06","slug":"integration-of-cosecx","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-of-cosecx\/","title":{"rendered":"Integration of Cosecx"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

Integration of Cosecx or Cosec x<\/h2>\n
\n

The integration of cosec x is log |cosec x – cot x| + C <\/strong>or \\(log |tan {x\\over 2}|\\) + C<\/strong>.<\/p>\n

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

i.e. \\(\\int\\) cosec x = log |cosec x – cot x| + C<\/p>\n

or, \\(\\int\\) cosec x = \\(log |tan {x\\over 2}|\\) + C<\/p>\n<\/blockquote>\n

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

\n

Let I = \\(\\int\\) cosec x dx.\u00a0<\/p>\n

Multiply and divide both denominator and numerator by cosec x – cot x.<\/p>\n

Then, I = \\(\\int\\) \\(cosec x(cosec x – cot x)\\over (cosec x – cot x)\\) dx<\/p>\n

Let cosec x – cot x = t. Then,<\/p>\n

d(cosec x – cot x) =dt<\/p>\n

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

\\(\\implies\\) dx = \\({dt\\over cosec x (cosec x – cot x)}\\)<\/p>\n

Putting cosec x – cot x = t and dx = \\({dt\\over sec x (sec x + tan x)}\\), we get<\/p>\n

I = \\(\\int\\) \\(sec x (sec x + tan x)\\over t\\) \\(\\times\\) \\({dt\\over cosec x (cosec x – cot x)}\\)<\/p>\n

= \\(\\int\\) \\(1\\over t\\) dt = log | t | + C<\/p>\n

= log |cosec x – cot x| + C<\/p>\n

Hence, I = log |cosec x – cot x| + C<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Evaluate \\(1\\over \\sqrt{1 – cos 2x}\\) dx.<\/p>\n

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

I = \\(1\\over \\sqrt{1 – cos 2x}\\)<\/p>\n

By using differentiation formula, 1 – cos 2x = \\(2 sin^2 x\\)<\/p>\n

\\(\\implies\\) I = \\(1\\over \\sqrt{2sin^2 x}\\)<\/p>\n

\\(\\implies\\) I = \\(1\\over \\sqrt{2}\\) \\(1\\over sin x\\) dx<\/p>\n

= \\(1\\over \\sqrt{2}\\) \\(\\int\\) cosec x dx<\/p>\n

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

\n

Related Questions<\/h3>\n

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

What is the Integration of Sec Inverse x and Cosec Inverse x ?<\/a><\/p>\n

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

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

Here you will learn proof of integration of cosecx or cosec x and examples based on it. Let’s begin – Integration of Cosecx or Cosec x The integration of cosec x is log |cosec x – cot x| + C or \\(log |tan {x\\over 2}|\\) + C. where C is the integration constant. i.e. \\(\\int\\) …<\/p>\n

Integration of Cosecx<\/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":[421,204,420],"yoast_head":"\nIntegration of Cosecx - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn proof of integration of cosecx or cosec 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-cosecx\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration of Cosecx - 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