Convergence of the integral of $1/|x|^p$ in $B(0,1)\subset\mathbb{R}^n$

Question

I'm looking for a reference to show the convergence for \begin{equation} \int_{B(0,1)} \frac{1}{|x|^p} \,\mathrm{d}x, \end{equation} where $B(0,1)\subset\mathbb{R}^n$ is the open unit ball, depending on $p$ and $n$.

Thank you.

Answer 1

Using spherical coordinates, one can see that the question of convergence of this integral is reduced to the question of convergence of the one dimensional integral

$$ \int_{0}^1 \frac{1}{r^p} r^{n-1} \, dr = \int_{0}^1 \frac{1}{r^{p - n + 1}} \, dr$$

as all the other trigonometric terms that result via the coordinate change don't affect the convergence. Thus, the integral converges if and only if $p - n + 1 < 1$ or $p < n$.