$f'(x).f(x)=x.\sin x\\ \displaystyle\int 2f'(x).f(x)\,dx=2\displaystyle\int x.\sin x\,dx\\ \Leftrightarrow f^2(x)=2\displaystyle\int x.\sin x\,dx=2I\\ I=\displaystyle\int x.\sin x\,dx\\ u=x \Rightarrow du=dx\\ dv=\sin x dx \Rightarrow v=-\cos x\\ I=-x\cos x+\displaystyle\int \cos x \, dx\\ =-x\cos x +\sin x+C\\ f(0)=\dfrac{\pi}{4}\\ \Rightarrow C=\dfrac{\pi^2}{32}\\ \Rightarrow f^2(x)=2I=-2x\cos x +2\sin x+\dfrac{\pi^2}{16}\\ f^2\left(\dfrac{\pi}{2}\right)=2+\dfrac{\pi^2}{16}$
