Average Speed for Maxwell's Distribution of Molecular Speed

[RaulTheUCSCSlug]

Using the Maxwell-Boltzmann equation above, there is an example in my book (Giancoli 4th edition p. 481) where they use this to find the average velocity. I understand that it would just be the sum of all the speeds of the molecules divided by the number of molecules. But then I'm having trouble on how they did the integration by parts? Could someone walk me through the process?

 

[mathman]

Let $a=\frac{m}{2kT}$
$\int_0^{\infty}v^3e^{-av^2}dv=\frac{1}{a}\int_0^{\infty}ve^{-av^2}dv=\frac{1}{2a^2}=\frac{2k^2T^2}{m^2}$

The first equality is integration by parts.

 

[phyzguy]

Do you know how to do integration by parts? $\int u dv = uv - \int v du$ . Try $u = x^2; dv = x e^{-x^2}dx$