Pointwise convergence of integral of Fourier series

[twizzy]

Homework Statement

If f(x) is a piecewise-continuous function in [-L,L], show that its indefinite integral F(x) = \int_{-L}^x f(s) ds has a full Fourier series that converges pointwise.

Homework Equations

Full Fourier series: f(x)=\frac{1}{2}A_0 + \sum_{n=1}^\infty A_n \cos (\frac{n \pi }{L}x) + B_n \sin (\frac{n \pi}{L}x)

Definition: \sum_{n=1}^\infty f_n (x) converges to f(x) pointwise in (a,b) if for each a<x<b we have
\Big| f(x) - \displaystyle{\sum_{n=1}^\infty f_n (x)} \Big| \to 0 as N\to\infty.

The Attempt at a Solution

I think I need to somehow justify integrating term-by-term, but am not sure how to proceed. Any ideas?

 

[ninty]

If you want to integrate term by term, you need uniform convergence.
Haven't really looked at this, so not saying that term by term integration is the solution here.