I want to study the convergence of the integral $$\displaystyle\int_{\mathbb{R}^n} (1+|\xi|^2)^{-s} d\xi,$$ where $s \in \mathbb{R}$. More precisely, I want to prove that the integral converges if $s > \frac{n}{2}$.
I tried using polar coordinates $\xi= r \cos \theta$, with $r >0$ and $\theta \in ]0,2\pi[$. Then we obtain $$ \displaystyle\int_{\mathbb{R}^n} (1+r^2|\cos(\theta)|^2)^{-s} dr d\theta $$ but I have no idea to how to continue.