$\int \frac{1}{\cos x \sqrt{\cos 2 x}} d x=$ (where C is a constant of integration.)

$\int \frac{1}{\cos x \sqrt{\cos 2 x}} d x=$ (where C is a constant of integration.)
  1. $\sin ^{-1}(\tan x)+C$
  2. $\log \left(\tan x+\sqrt{\tan ^2 x+1}\right)+C$
  3. $\tan ^{-1} x+C$
  4. $\log \left(\tan x+\sqrt{\tan ^2 x-1}\right)+C$

Solution

$\begin{aligned} & \int \frac{d x}{\cos x \sqrt{\cos 2 x}}=\int \frac{d x}{\cos x \sqrt{\frac{1-\tan ^2 x}{1+\tan ^2 x}}} \\ & \int \frac{d x}{\cos x \frac{\sqrt{1-\tan ^2 x}}{\sec x}}=\int \frac{\sec ^2 d x}{\sqrt{1-\tan ^2 x}} \\ & \int \frac{d t}{\sqrt{1-t^2}}=\sin ^{-1}(t)+C=\sin ^{-1}(\tan x)+C\end{aligned}$ [let $\tan \mathrm{x}=\mathrm{t}$ ]

Asked in: MHT CET 2022 (06 Aug Shift 1)

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