\begin{equation} \bar{v}=\int_{0}^{\infty}vG(v)dv=4\pi\left(\frac{m}{2\pi kT}\right)^{3 /2}\int_{0}^{\infty}e^{-mv^2/2kT}v^3dv \end{equation} Using the integral $\int_{0}^{\infty}x^{2n+ 1}e^{-ax^2}dx=\frac{n!}{2a^{n+1}}$ \begin{equation} \bar{v}=4\pi\left(\frac{m} {2\pi kT}\right)^{3/2}\frac{1}{2(m/2kT)^2}=\left(\frac{8kT}{\pi m}\right)^{1 /2} \end{equation} Multiplying and dividing by Avogadro's number and taking into account that $kN_A=R$ and $N_A m=M$ \begin{equation} \bar{v}=\left(\frac{ 8RT}{\pi M}\right)^{1/2} \end{equation}