Question $1$: Use $F(v)dv$ function for $v$ to find $($:$v^3$:$)$ for ideal gas molecule $($show $($:$v^3$:$)$ in terms of $($:$v$:$).$

$F($:$v$:$)dv=4v^2(\frac{m}{2kT}{)}^{\frac{3}{2}}$exp$(-\frac{m{v}^2}{2kT})$

$($:$v$:$)=(\frac{8kT}{m}{)}^{\frac{1}{2}}$

Definite Integrals Occurring in the Kinetic Theory of Gases

$\backslash$table$[[,n,],[$Integral$,0,1,2,3,4,5],[{\displaystyle \underset{0}{\overset{}{}}}{x}^n$exp$(-a{x}^2)dx,\frac{1}{2}(\frac{}{a}{)}^{\frac{1}{2}},\frac{1}{2a},\frac{1}{4}(\frac{}{a^3}{)}^{\frac{1}{2}},\frac{1}{2a^2},\frac{3}{8}(\frac{}{a^5}{)}^{\frac{1}{2}},\frac{1}{a^3}$