I've got this thing on my mind : we know that $cos(x)$ is a periodic function , hence integral from $2(k-1) \pi$ to $2k \pi$ will yield the same value for any $k \geq1$.
My question is , why is this integral property still true when considering $ \cos^{n} (x)$ ? Intuitively , it seems right , but I would be much more comfortable with a rigorous explanation .
Thanks for your patience!
As Panda Bear notes in a comment, since $\cos(x)$ is periodic with period $2\pi$, we have $\cos(x+2\pi)=\cos(x)$ for any value of $x$. Thus, $\cos^n(x+2\pi)=\cos^n(x)$ for any value of $x$ and $\cos^n(x)$ is also periodic with period $2\pi$.