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**Opalg** If you think in terms of computing the integral using a spherical shell method, you get

$\displaystyle \int_{\mathbb{R}^n} \frac{1}{(1 + |x|^2)^N} \, dx = \int_0^\infty \frac{1}{(1 + r^2)^N}S_{n-1}(r)\,dr,$

where $\displaystyle S_{n-1}(r)$ is the "surface area" (or more correctly the (n–1)-dimensional volume) of the (n–1)-sphere. Since $\displaystyle S_{n-1}(r) = k_nr^{n-1}$ for some constant $\displaystyle k_n$, it follows that $\displaystyle \int_{\mathbb{R}^n} \frac{1}{(1 + |x|^2)^N} \, dx = k_n\int_0^\infty \frac{r^{n-1}}{(1 + r^2)^N}\,dr,$ and that converges for $\displaystyle N\geqslant\tfrac12(n+1)$ by comparison with the integral of $\displaystyle r^{-2}.$