For a Fourier method problem, I need to solve the following integral for the closed forms of coefficients

For n even, the integral is 0 since sin(nx) would be odd about

For n odd, I let n = 2m + 1 and hence my integral with the sin(a+b) identity gives

but here I'm not sure what to do next. perhaps there is an easier way to approach this problem. any suggestions would be greatly appreciated.