Derive the mean speed, most probable speed and the root mean square speed

from the Maxwell distribution, and calculate the each speed of $N_2$ in air at $298$ K

$($ref$.$ Example $20\mathrm{.}1$ in p$750)$

${\displaystyle \underset{0}{\overset{}{}}}{x}^{2n}{e}^{-a{x}^2}dx=\frac{1*3\text{c}\text{d}\text{o}\text{t}\text{s}(2n-1)}{2^{n+1}*a^n}*(\frac{}{a}{)}^{\frac{1}{2}}$

${\displaystyle \underset{0}{\overset{}{}}}{x}^3{e}^{-a{x}^2}dx=\frac{1}{2a^2}$