Question

$\int \frac{dx}{\cos x\sqrt{\cos 2x}}=\frac{1}{m}{\sin }^{-1}\left(\tan x\right)$. Find $m$

Answer 1

$\int \frac{dx}{\cos x\sqrt{\cos 2x}}$

$\int \frac{dx}{\cos x\sqrt{{\cos }^{2}x-{\sin }^{2}x}}$

$\int \frac{dx}{\cos x\cos x\sqrt{1-\frac{{\sin }^{2}x}{{\cos }^{2}x}}}$

$\int \frac{dx}{{\cos }^{2}x\sqrt{1-{\tan }^{2}x}}$

$\int \frac{{\sec }^{2}xdx}{\sqrt{1-{\tan }^{2}x}}$

put $\tan x=t$

${\sec }^{2}xdx=dt$

$\int \frac{dt}{\sqrt{1^2-t^2}}={\sin }^{-1}t+c$

$={\sin }^{-1}\left(\tan x\right)+c$