Answer
$$\int_{0}^{\pi}x\,sin\,xcos\,x\,dx=-\frac{\pi}{4}$$
Work Step by Step
$$\int_{0}^{\pi}x\,sin\,xcos\,x\,dx=\frac{1}{2}\int_{0}^{\pi}x\,sin\,2x\,dx$$
$$\frac{1}{2}\left[\left[-\frac{x\,cos\,2x}{2}\right]_{0}^{\pi}+\frac{1}{2}\int_{0}^{\pi}cos\,2x\,dx\right]$$
$$=-\frac{\pi}{4}+\frac{1}{4}\left[\frac{sin\,2x}{2}\right]_{0}^{\pi}$$
$$=-\frac{\pi}{4}$$
