• Support PF! Buy your school textbooks, materials and every day products Here!

Tricky integral problem

  • Thread starter ppyadof
  • Start date
  • #1
41
0

Homework Statement


This is the problem: I need to prove from the Maxwell-Boltzmann distributions the expression for the rms speed.


Homework Equations


[tex] v_{rms} = \sqrt{\frac{3kT}{m}} [/tex]
[tex] < v^2 > = 4\pi(\frac{m}{2\pi kT})^{3/2} \int v^4 e^{\frac{-mv^2}{2kT}} dv [/tex]
The limits on the integral are infinity and 0.

From what I know;
[tex] v_{rms} = sqrt{ <v^2> } [/tex]

The Attempt at a Solution


I did a similar one like this to prove the corresponding expression for <v>. I used a substitution and then used parts on the resulting expression, but if use that method on this one, you get a factor of [itex] u^{3/2} [/itex] times by the exponential (assuming the substitution is [itex] u = v^2 [/itex]). This obviously causes problems, so I was wondering is there another way to do this proof without resorting to an integral or is there an easy way of doing this integral?

P.S. Sorry if this thread should be in the advanced physics part. I only thought of that after I posted.
 
Last edited:

Answers and Replies

  • #2
George Jones
Staff Emeritus
Science Advisor
Gold Member
7,273
808
Do you know how to calculate (via the standard trick)

[tex]\int_0^\infty e^{-a x^2} dx?[/tex]

If you do, then using integration by parts a couple of times will reduce your integral to an integral of this form.
 
  • #3
AKG
Science Advisor
Homework Helper
2,565
3
1. Do a change of variable just to neaten up the expression (i.e. absorb the constants into your variable with respect to which you're integrating so that you can take all constants out of the integral)
2. Use integration by parts
3. Fact:

[tex]\int _{-\infty} ^{\infty}e^{-x^2}dx = \sqrt{\pi }[/tex]
 
  • #4
dextercioby
Science Advisor
Homework Helper
Insights Author
12,981
540
Do you know how to calculate (via the standard trick)

[tex]\int_0^\infty e^{-a x^2} dx?[/tex]

If you do, then using integration by parts a couple of times will reduce your integral to an integral of this form.
It's nicer if you treat "a" as a parameter and consider the second derivative of the result...:wink:
 

Related Threads on Tricky integral problem

  • Last Post
Replies
2
Views
923
  • Last Post
Replies
5
Views
1K
Replies
2
Views
2K
Replies
11
Views
2K
Replies
8
Views
1K
Replies
2
Views
560
  • Last Post
Replies
1
Views
844
  • Last Post
Replies
9
Views
1K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
1
Views
924
Top