For fourier series g(x), prove that the fourier series for the integral G(x) can be found by term-by-term integration of g(x)

Question

I want to prove that if I have a fourier series of the form $g(x) = a_0/2 + {\sum_i}^\infty a_icos(ix) + b_isin(ix) $, the fourier series of G(x) $-x*a_0/2$ can be found by simply integrating g(x) term by term.

I've found the term by term integration of g(x), but I'm confused by how I can know that this is in fact the fourier series for $G(x) -x*a_0/2$.