1. The problem statement, all variables and given/known data Let f : R → Rn be a smooth function. Give necessary and sufficient conditions on f so that the antiderivative F(x) = ∫f(t)dt (from 0 to x) is periodic with period p ≠ 0 2. Relevant equations 3. The attempt at a solution My initial thought is that as long as f is periodic then F will be periodic as well -- the oscillatory nature of the differentiation operators on Cos and Sin comes to mind. But I imagine it can't be that simple, perhaps there's extra conditions which are stipulated to ensure F is periodic. Or maybe my thought that f need be periodic is not strictly true. Any guidance?