Suppose x is a real vector in n dimensions. What is the general formula for the result of the following integral, which is calculated over the entire space:
$$ I = \int\limits_{{\mathbb{R}^n}} {{e^{ - ||{\bf{x}}||}}} {d^n}{\bf{x}}\ $$
Where ||x|| is L2-norm of x.
In any dimension you can use the (hyper-)spherical coordinates, in which $d^n\mathbf{x}=r^{n-1}drd\Omega$, and $\Omega$ is the generalization of solid angle. In this coordinate system your integral becomes $$ I_n=\int d\Omega\int_{0}^\infty r^{n-1} e^{-r}dr $$ The latter integral, by definition, is $\Gamma(n)=(n-1)!$, so it remains to calculate $\int d\Omega$ (oh by the way by this integral I mean integral over all solid angles). Let us denote this integral by $\sigma_{n}$.
Note that $$J=\int_{\mathbb{R}^n} e^{-||\mathbf{x}||^2}d^n\mathbf{x}=\left(\int_{-\infty}^\infty e^{-x^2}dx\right)^n =\pi^{\frac{n}{2}}$$ But one can calculate the same integral in (hyper-)spherical coordinates, then $$ J=\sigma_{n}\int_{0}^\infty r^{n-1}e^{-r^2}dr\xrightarrow{s=r^2} \frac{\sigma_n}{2}\int_{0}^\infty s^{(\frac{n}{2}-1)}e^{-s}ds=\frac{\Gamma(\frac{n}{2})}{2}\sigma_n\Longrightarrow \boxed{\sigma_{n}=\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}} $$ As a result your integral is $$ \boxed{I_n=2\pi^{\frac{n}{2}}\frac{\Gamma(n)}{\Gamma(\frac{n}{2})}} $$