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  4. ∫ sec 3x dx is equal to

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Q. $ \int{{{\sec }^{3}}x}\,\,dx $ is equal to

Jharkhand CECEJharkhand CECE 2010

Solution:

Let $ I=\int{{{\sec }^{2}}x}\,\,dx $
$ \Rightarrow $ $ I=\int{\sec x}\cdot {{\sec }^{2}}x\,\,dx $
$ \Rightarrow $ $ I=\sec x\tan x-\int{\sec x\tan x\cdot \tan x\,\,dx} $ (integrating by parts) $ I=\sec x\tan x-\int{\sec x}{{\tan }^{2}}x\,\,dx $
$ =\sec x\tan x-\int{\sec x({{\sec }^{2}}x-1)dx} $
$ \Rightarrow $ $ I=\sec x\tan x-\int{({{\sec }^{2}}x-\sec x)dx} $
$ \Rightarrow $ $ I=\sec x\tan x-\int{{{\sec }^{2}}x\,\,dx+\int{\sec x\,\,dx}} $
$ \Rightarrow $ $ 2I=\sec x\tan x+\int{\sec x\,\,dx} $
$ \Rightarrow $ $ 2I=\sec x\tan x+\log |\sec x+\tan x|+c $
$ \Rightarrow $ $ I=\frac{1}{2}\sec x\tan x+\frac{1}{2}\log |\sec x+\tan x|+c $