Question

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

Answer 1

$I=\int \frac{\cos x}{\sqrt{\cos 2x}}dx=\int \frac{\cos x}{\sqrt{\cos {}^{2}x\sin {}^{2}x}}dxI=\int \frac{\cos x}{\sqrt{1-2\sin {}^{2}x}}dx\sin x=t,\cos xdx=dt\therefore I=\int \frac{dt}{\sqrt{1-2t^2}}=\int \frac{dt}{\sqrt{1-{\left(\sqrt{2t}\right)}^2}}={\sin }^{-1}\left(\sqrt{2}t\right)+C={\sin }^{-1}\left(\sqrt{2}\sin x\right)+C$