Definite Integral of Periodic Function

Theorem
Let $f$ be a Darboux integrable periodic function with period $L$.

Let $\alpha \in \R$ and $n \in \Z$.

Then:
 * $\displaystyle \int_\alpha^{\alpha + n L} \map f x \d x = n \int_0^L \map f x \d x$

Proof
For $n \ge 0$:

For $n < 0$:

Hence the result.