Primitive of Periodic Real Function

Theorem
Let $f: \mathbb F \to \mathbb F$ be a function, where $\mathbb F \in \left\{{\R, \C}\right\}$.

Let $f$ have a primitive.

If $f$ is periodic with period $L$, then any primitive of $f$ is also periodic with period $L$.

Proof
Let $f$ be periodic with period $L$.

Let $f$ have a primitive $F$.

Then:

When $C = 0$, $F$ is periodic by definition.

For all other $C$, we invoke Periodic Function plus Constant.

Also see

 * Derivative of Periodic Function