Let $f(x)=\sin(x)+x\cos(x)$, $x\in(-\pi,\pi)$.
- Find the Fourier series of $f$
I know all the formulas for the coefficients $a_0,b_0,c_0$ and $a_n,b_n,c_n$ and the property for odd functions that then $c_k(f)=c_{-k}(f)$ i.e $a_k(f)=0$ and that we can write it as a sine series$$\sum_{k=1}^{\infty}b_k(f)\sin(kx)$$
Is there a trick I am oveseeing as brute forcing it seems really tough and pages long?
