The following integrals are the Brasilian Mathematical Olympiad in 2019.
Show that $\displaystyle\int_1^2\frac{e^x(x-1)}{x(x+e^x)}\,dx=\ln\left(\frac{2+e^2}{2+2e}\right)$ and $\displaystyle\int_0^{\pi/2}\frac{x\sin(2x)}{1+\cos(2x)^2}\,dx=\frac{\pi^2}{16}$.
I tried to use traditional methods, but these integrals are very difficult. Has someone any suggestion? Thanks a lot.