Integration in $\mathbb{R}^n$ similar to arctan(x)

Question

Suppose we have $ x,y \in \mathbb{R}^n.$ let $||x||$ be the euclidean norm. How does one calculate

$\int_{||y||>||x||} \frac{dy}{1+||y||^2}$

Does it converge ?

Answer 1

No, even in $\Bbb R^2$ it diverges, and matters only get worse as $n$ gets bigger.

In polar coordinates, we'll have $$\int_{r\ge R} \frac{r}{1+r^2}\,dr\,d\theta,$$ which diverges. In higher dimensions, spherical coordinates will give a constant times $$\int_{r\ge R}\frac{r^{n-1}}{1+r^2}\,dr,$$ which is even worse. Sorry.