Antiderivative of periodic function I practise before the real analysis exam and I got stuck on few excercises.
"if $F$ is an antiderivative of a continuous periodic function and $F$ is bounded, then $F$ is periodic"
$~\to~$ one should give a proof or counterexample.
(I'm convinced it's true, but how to prove it?)

Hint:
What is special about $1 + \sin(x)$ that gives it the following property?
$$\lim_{R\to\infty} \int_0^{R}1 + \sin(x) \ dx = \infty$$
If this integral was to have a maximum, what points would you look for first?
Further, $$\frac{d}{dR}\int_0^{R}f(x) dx = f(R)$$
