Suppose $f$ is $2\pi$-peridic and integrable on any finite interval. 

Prove that if $a,b \in \mathbb{R}$, then $$
\int\_{a}^{b} f(x)\ dx=\int\_{a+2\pi}^{b+2\pi} f(x)\ dx =\int\_{a-2\pi}^{b-2\pi}f(x)\ dx . $$
Also prove that $$\int\_{-\pi}^{\pi} f(x+a)\ dx=\int\_{-\pi}^{\pi} f(x)\ dx =\int\_{-\pi +a}^{\pi+a}f(x)\ dx . $$