Fact: Using the substitution $u = \pi - x,$ one can show that $$\int_{0}^{\,\pi} {xf(\sin x)}\, dx = \frac{\pi}{2} \int_{0}^{\,\pi} {f(\sin x)}\, dx.$$

Use the fact given above to evaluate the integral $$\int_{0}^{\,\pi} {\frac{3 x\sin(x)}{1+\cos^2(x)}}\, dx$$