Change of variables in a simple integral

In summary, the conversation discusses the issue of whether the power in the second integral should be 1/2 or 3/2. The person arguing for 1/2 cites a source that supports their claim, while the other person suggests checking the dimension to verify the correct power.
  • #1
MathematicalPhysicist
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Homework Statement
I have the integral ##\int_0^\infty dv v^2 x \frac{\partial g}{\partial x}##.
Using ##x=\beta(\frac{1}{2}mv^2-\mu)## we get:
##\int_{-\infty}^\infty dx x (x+\beta \mu)^{3/2}\frac{\partial g}{\partial x}##.

The last integral upto some constants which don't get integrated.
Relevant Equations
they appear already in the HW statement.
So we have ##x=\beta(1/2 mv^2-\mu)##, i.e ##\sqrt{2(x/\beta+\mu)/m}=v##.
##dv= \sqrt{2/m}dx/\sqrt{2(x/\beta+\mu)/m}##.

So should I get in the second integral ##(x+\beta \mu)^{1/2}##, since we have: $$v^2 dv = (2(x/\beta+\mu)/m)\sqrt{2/m} dx/\sqrt{2(x/\beta+\mu)/m}$$

So shouldn't it be a power of 1/2 and not 3/2?

Thanks in advance!
 
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  • #2
You show no explanation about ##\frac{\partial g}{\partial x}##.
 
  • #3
anuttarasammyak said:
You show no explanation about ##\frac{\partial g}{\partial x}##.
Well, it's written that ##x## and ##g## both depend on ##v^2## only.
How does that fact change any thing here?
 
  • #4
Ah, besides the fact that ##\partial g/\partial x = \partial g /\partial v \partial v / \partial x##, but I am not sure he changes it here that way...
 
  • #5
I believe it should be 1/2 and not 3/2 because in the SM of Reif on equation (8) of page 108 he changes the LHS to 1/2 instead of 3/2.

Reading the SM alongside the problems and trying to understand the solutions can be quite time consuming, but hey that's what I chose to do. :-)
 
  • #6
How about checking it by looking at dimension ? Apart from ##\frac{\partial g}{\partial x}## integral has dimension of ##L^4T^{-3}## if it is correct to take v and x velocity and coordinate.
 

1. What is a change of variables in a simple integral?

A change of variables in a simple integral is a method used to transform an integral into a different form, making it easier to solve. This is done by substituting a new variable in place of the original variable in the integral.

2. Why is a change of variables useful in integrals?

A change of variables can make an integral easier to solve by simplifying the integrand or making it possible to use known integration techniques. It can also help to find the bounds of integration in more complex integrals.

3. What are the steps for performing a change of variables in a simple integral?

The steps for performing a change of variables in a simple integral are:

  1. Choose a new variable to substitute in the integral.
  2. Find the relationship between the new variable and the original variable.
  3. Substitute the new variable and its relationship in the integral.
  4. Simplify the integral and solve for the new variable.
  5. Substitute the original variable back in the solution to get the final answer.

4. What are some common substitution choices for a change of variables in a simple integral?

Some common substitution choices for a change of variables in a simple integral include trigonometric functions, exponential functions, and inverse functions. The choice of substitution depends on the integrand and the goal of simplifying the integral.

5. Are there any precautions to take when using a change of variables in a simple integral?

Yes, there are a few precautions to take when using a change of variables in a simple integral. It is important to make sure that the new variable and its relationship are valid for the given integral. Also, the bounds of integration may need to be adjusted when substituting a new variable, so it is important to carefully consider the new bounds. Additionally, it is important to check the final answer by substituting the original variable back in to ensure the correct solution is obtained.

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