I am trying to carry out this integration but I seem to be going wrong:
$$I=\int_{0}^{\frac{\pi}{2}}\frac{x dx}{\sin x+\cos x}=\int_{0}^{\frac{\pi}{2}}\frac{(\frac{\pi}{2}-x) dx}{\sin(\frac{\pi}{2}-x)+\cos (\frac{\pi}{2}-x)} \implies 2I=\frac{\pi}{2}\int_{0}^{\frac{\pi}{2}}\frac{ dx}{\sin x+\cos x}$$
I am not able to proceed from here.