Chapter 8 - Integration Techniques, L'Hopital's Rule, and Improper Integrals - Review Exercises - Page 580: 65

$$\pi $$

$$\eqalign{ & \int_0^\pi {x\sin x} dx \cr & {\text{Integrate by tables, use }}\int {u\sin udu} = \sin u - u\cos u + C \cr & \int_0^\pi {x\sin x} dx = \left[ {\sin x - x\cos x} \right]_0^\pi \cr & {\text{Evaluate}} \cr & = \left[ {\sin \pi - \pi \cos \pi } \right] - \left[ {\sin 0 - 0\cos 0} \right] \cr & = 0 - \pi \left( { - 1} \right) - 0 \cr & = \pi \cr} $$