The average speed of molecules in an ideal gas is $$\overline{v} = \frac{4}{\sqrt{\pi}}\left(\frac{M}{2RT}\right)^{3/2} \int_{0}^{\,\infty} {v^3 e^{-Mv^2/(2RT)}}\, dv$$ where $M$ is the molecular weight of the gas, $R$ is the gas constant, $T$ is the gas temperature, and $v$ is the molecular speed. Determine an expression for $\overline{v}$ in terms of $M$, $R$, and $T$ by computing the improper integral and simplifying the result.

$\overline{v} =$