{"id":5824,"date":"2021-10-03T17:52:23","date_gmt":"2021-10-03T12:22:23","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5824"},"modified":"2021-11-26T00:00:09","modified_gmt":"2021-11-25T18:30:09","slug":"integration-of-tanx","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-of-tanx\/","title":{"rendered":"Integration of Tanx"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

Integration of Tanx or Tan x<\/h2>\n
\n

The integration of tanx is – log |cos x| + C<\/strong> or log |sec x| + C<\/strong><\/p>\n

i.e. \\(\\int\\) (tanx) dx = – log |cos x| + C or,<\/p>\n

\\(\\int\\) (tanx) dx = log |sec x| + C<\/p>\n<\/blockquote>\n

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

\n

Let I = \\(\\int\\) (tan x) dx<\/p>\n

Then, I = \\(\\int\\) \\(sin x\\over cos x\\) dx<\/p>\n

Let cos x = t\u00a0<\/p>\n

Then, d(cos x) = dt \\(\\implies\\) -sin x dx = dt\u00a0<\/p>\n

\\(\\implies\\) dx = \\(-dt\\over sin x\\)<\/p>\n

Putting cos x = t, and dx = \\(-dt\\over sin x\\), we get<\/p>\n

I = \\(\\int\\) \\(sin x\\over cos x\\) \\(\\times\\) \\(-dt\\over sin x\\)<\/p>\n

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

= – log |cos x| + C<\/p>\n

And cos x = \\(1\\over sec x\\)<\/p>\n

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

Hence, \\(\\int\\) (tanx) dx = – log |cos x| + C or, \\(\\int\\) (tanx) dx = log |sec x| + C<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Evaluate : \\(\\int\\)\u00a0\\(\\sqrt{{1-cos 2x}\\over {1+cos 2x}}\\) dx<\/p>\n

Solution<\/span><\/strong> : We have,\u00a0<\/p>\n

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

By Trigonometry formulas,<\/p>\n

1 – cos 2x = \\(2sin^2 x\\) and 1 + cos 2x = \\(2cos^2 x\\)<\/p>\n

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

\\(\\implies\\) I = \\(\\int\\) \\({sin x}\\over {cos x}\\) dx<\/p>\n

{\\(\\because\\) \\({sin x}\\over {cos x}\\) = tan x }<\/p>\n

\\(\\implies\\) I = \\(\\int\\) tan x dx\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/p>\n

\\(\\implies\\) I = log |sec x| + C = – log |cos x| + C<\/p>\n

\n

Related Questions<\/h3>\n

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

What is the Integration of tan inverse x ?<\/a><\/p>\n

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

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

Here you will learn proof of integration of tanx or tan x and examples based on it. Let’s begin – Integration of Tanx or Tan x The integration of tanx is – log |cos x| + C or log |sec x| + C i.e. \\(\\int\\) (tanx) dx = – log |cos x| + C or, …<\/p>\n

Integration of Tanx<\/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":[427,204,426],"yoast_head":"\nIntegration of Tanx - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn proof of integration of tanx or tan 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-tanx\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration of Tanx - 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