Chapter 6 - Applications of the Integral - 6.2 Setting Up Integrals: Volume, Density, Average Value - Exercises - Page 298: 48

$\frac{1}{n\pi }(1-\cos (n\pi)).$

The average is given by $$ \frac{1}{\pi-0} \int_{0}^{\pi} \sin(nx) d x=\frac{1}{n\pi } \int_{0}^{\pi} n\sin(nx) d x\\=-\left.\frac{1}{n\pi } \cos(nx) \right|_{0} ^{\pi}=\frac{1}{n\pi }(1-\cos (n\pi)). $$