\[\int_0^\infty e^{-\pi r^2} r^{n-1} {\rm d} r = \frac{1}{2\pi^{n/2}} \cdot \int_0^{\infty} e^{-u} u^{n/2-1} {\rm d} u = \frac{\Gamma(n/2)}{2\pi^{n/2}}\; ,\]