Calculate $\int_{\mathbb R^n}\frac{1}{(1+\|x\|_2^n)^2}dx$

Question

How can I calculate the integral
$$ \int_{\mathbb R^n}\frac{1}{(1+\|x\|_2^n)^2}dx $$? Is there a "simple" way?

Answer 1

With $r=\|x\|_2$ the integral becomes \begin{align} \int_{ \mathbb R^n }\frac{1}{ (1+\|x\|_2^n)^2 }dx &= \int_\Omega\int_{r=0}^\infty\frac{1}{(1+r^n)^2}r^{n-1} dr d\Omega \\ &= \Omega\int_{r=0}^\infty\frac{1}{(1+r^n)^2}r^{n-1}dr \\ &= \Omega\left( \frac{-1/n}{1 + r^n} \right)\Bigg|_{r=0}^\infty \\ &= \frac{\Omega}{n} \end{align} where volume form is written as $dx = r^{n-1} dr d \Omega$, and $\Omega$ is the total solid angle in $\mathbb R^n$ and $d\Omega$ is its differential element. I don't remeber what this $\Omega$ is but you can find it in Wiki.