Evaluate $\int_0^\frac{\pi}{4}\frac{\sin x}{\cos x\sqrt{\cos 2x}} \,\rm{d}x$ 
Evaluate $$\int_0^\frac{\pi}{4}\frac{\sin x}{\cos x\sqrt{\cos 2x}} \,\rm{d}x$$

I was thinking to write $\cos 2x = \cos^2 x - \sin^2 x$ and then to factor out $\sin^2x$ to try to simplify the expression, but gets me nowhere. Any help will be appreciated!

\begin{align}J&=\int_0^\frac{\pi}{4}\frac{\sin x}{\cos x\sqrt{\cos 2x}} \,\rm{d}x\\
&=\int_0^\frac{\pi}{4}\frac{\sin x}{\cos x\sqrt{2\cos^2 x-1}} \,\rm{d}x\\
&\overset{t=\cos x}=\int_{\frac{1}{\sqrt{2}}}^1\frac{1}{x\sqrt{2t^2-1}}\,dt\\
&\overset{x=t^2}=\frac{1}{2}\int_{\frac{1}{2}}^1\frac{1}{x\sqrt{2x-1}}\,dx\\
&\overset{y=\sqrt{2x-1}}=\int_0^1 \frac{1}{1+y^2}\,dy\\
&=\arctan(1)-\arctan(0)\\
&=\boxed{\frac{\pi}{4}}
\end{align}
