These integrals are often used in solving problems in Statistical Mechanics.
$ \int_0^{\infty} e^{-Bv^2} v^n dv = \frac{1}{2B^{(\frac{n+1}{2})}} \Gamma (\frac{n+1}{2}) $, where $ \Gamma $ is gamma function
$ \Gamma(\frac{1}{2}) = \sqrt{\pi}$
$ \Gamma(1) = 1$
$ \Gamma(\frac{3}{2}) =\frac{ \sqrt{\pi}}{2}$
$ \Gamma(2) = 1$