Doing definite integration $\int_0^{\pi/4}\frac{x\,dx}{\cos x(\cos x + \sin x)}$ We have to solve the following integration
$$
\int_0^{\pi/4}\frac{x\,dx}{\cos x(\cos x + \sin x)}
$$
I divided both in Nr and Dr by $\cos^2 x$.
But after that I stuck.

We have $$I=\int_{0}^{\pi/4}\frac{2x}{2\cos^2x+2\cos x\sin x}dx=\int_0^{\pi/4}\frac{2x}{\cos 2x+\sin 2x+1}dx=\frac{1}{2}\int_0^{\pi/2}\frac{xdx}{\cos x+\sin x+1}=\frac{1}{2}\int_0^{\pi/2}\frac{\pi/2-x}{\cos x+\sin x+1}dx\\I=\frac{1}{4}\int_0^{\pi/2}\frac{\pi/2}{\cos x+\sin x+1}dx$$Now substitute $x$ for $2x$ again$$=\frac{\pi}{4}\int_0^{\pi/4}\frac{dx}{2\cos^2x+2\sin x\cos x}=\frac{\pi}{8}\int_0^{\pi/4}\frac{\sec^2xdx}{1+\tan x}=\frac{\pi}{8}\int_0^{1}\frac{1}{1+u}du=\frac{\pi}{8}\log 2$$
