2.4 Exercise 35

Verify the normalization equation $\int_{0}^{\infty} f \left(v\right) d v = 1 .$ In doing the integral, first make the substitution $u = \sqrt{\frac{m}{2 k_{\text{B}} T}} v = \frac{v}{v_{p}} .$ This “scaling” transformation gives you all features of the answer except for the integral, which is a dimensionless numerical factor. You’ll need the formula

$\int_{0}^{\infty} x^{2} e^{− x^{2}} d x = \frac{\sqrt{\pi}}{4}$

to find the numerical factor and verify the normalization.