Chapter 8.2, Problem 97E

Vibrating String A string stretched between the two points (0, 0) and (2, 0) is plucked by displacing the string h units at its midpoint. The motion of the string is modeled by a Fourier Sine Series whose coefficients are given by

$b_n=h{\displaystyle \int x\sin \frac{n\pi x}{2}dx+h{\displaystyle {\int }_1^2(-x+2)\sin \frac{n\pi x}{2}dx}}$

Find $b_n$.