{"id":5831,"date":"2021-10-03T18:34:48","date_gmt":"2021-10-03T13:04:48","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5831"},"modified":"2021-11-25T23:56:00","modified_gmt":"2021-11-25T18:26:00","slug":"integration-of-secx","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-of-secx\/","title":{"rendered":"Integration of Secx"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

Integration of Secx or Sec x<\/h2>\n
\n

The integration of sec x is log |sec x + tan x| + C\u00a0<\/strong>or \\(log |tan ({\\pi\\over 4} + {x\\over 2})|\\) + C<\/strong>.<\/p>\n

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

i.e. \\(\\int\\) sec x = log |sec x + tan x| + C<\/p>\n

or, \\(\\int\\) sec x = \\(log |tan ({\\pi\\over 4} + {x\\over 2})|\\) + C<\/p>\n<\/blockquote>\n

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

\n

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

Multiply and divide both denominator and numerator by sec x + tan x.<\/p>\n

Then, I = \\(\\int\\) \\(sec x(sec x + tan x)\\over (sec x + tan x)\\) dx<\/p>\n

Let sec x + tan x = t. Then,<\/p>\n

d(sec x + tan x) =dt<\/p>\n

\\(\\implies\\) \\((sec x tan x + sec^2 x)\\) dx = dt<\/p>\n

\\(\\implies\\) dx = \\({dt\\over sec x (sec x + tan x)}\\)<\/p>\n

Putting sec x + tan 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 sec x (sec x + tan x)}\\)<\/p>\n

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

= log |sec x + tan x| + C<\/p>\n

Hence, I = log |sec x + tan x| + C<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : 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 cos^2 x\\)<\/p>\n

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

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

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

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

\n

Related Questions<\/h3>\n

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

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

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

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

Here you will learn proof of integration of secx or sec x and examples based on it. Let’s begin – Integration of Secx or Sec x The integration of sec x is log |sec x + tan x| + C\u00a0or \\(log |tan ({\\pi\\over 4} + {x\\over 2})|\\) + C. where C is the integration constant. …<\/p>\n

Integration of Secx<\/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":[423,204,422],"yoast_head":"\nIntegration of Secx - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn proof of integration of secx or sec 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-secx\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration of Secx - 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