Tricky integral problem

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In summary, the problem is to prove the expression for the rms speed from the Maxwell-Boltzmann distributions using the equations v_{rms} = \sqrt{\frac{3kT}{m}} and < v^2 > = 4\pi(\frac{m}{2\pi kT})^{3/2} \int v^4 e^{\frac{-mv^2}{2kT}} dv with the limits on the integral being infinity and 0. The poster attempted to use a substitution and integration by parts, but ran into difficulties. They asked for an alternative method or a way to simplify the integral. Another poster suggested using the standard trick for calculating \int_0^\infty e^{-a
  • #1
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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.
 
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  • #2
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
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
George Jones said:
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:
 

1) What is a tricky integral problem?

A tricky integral problem refers to a mathematical integration question that is considered difficult or challenging to solve. It often requires advanced mathematical knowledge and problem-solving skills to arrive at the correct solution.

2) How do I approach a tricky integral problem?

It is important to first understand the concept and properties of integrals, and then carefully analyze the given problem. You can use various techniques such as substitution, integration by parts, or trigonometric identities to simplify the problem and make it easier to solve.

3) What are some common mistakes when solving a tricky integral problem?

Some common mistakes include forgetting to apply the correct integration rules, making errors in algebraic manipulation, and overlooking the limits of integration. It is crucial to double-check your work and be mindful of these potential mistakes.

4) Are there any tips for solving tricky integral problems?

One tip is to break down the problem into smaller parts and solve each part separately. Also, try to visualize the problem and use geometric intuition to guide your thinking. Additionally, practicing and familiarizing yourself with different integration techniques can help you approach tricky problems with confidence.

5) How can I check if my solution to a tricky integral problem is correct?

You can always verify your solution by differentiating it and seeing if it matches the original function. Another way is to use online integral calculators or ask a fellow mathematician or professor to review your work. It is also helpful to solve the problem using different methods to confirm your solution.

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