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Homework Help: Change of Variables

  1. Apr 6, 2015 #1
    1. The problem statement, all variables and given/known data
    In the book it gave the example for standard change of variables as,
    z = x / 1 + x or equivalently x = z / 1 - z , then

    dx = dz / (1 - z)^2 , thus (2)

    1 / (1 - z)^2 f (z / 1 - z) dz (3)

    This is what I am trying to accomplish but with the expression x^3 / (e^x - 1) dx. So I can put this expression equal to z and find equivalent equal to x then find dx eq(2) then final result eq (3).

    2. Relevant equations
    integral of f(x) dx

    3. The attempt at a solution
    I found the derivative to be (3*x^2/(e^x - 1)) - (x^3 * e^x / (e^x - 1)^2) but not getting the final result like in equation (3) from above maybe I am doing change of variables wrong. I have seen the formulas with variables online but still not getting it. If someone could help that would be GREAT!
  2. jcsd
  3. Apr 7, 2015 #2


    User Avatar
    Gold Member

    Sorry you're just now getting a response. Welcome to PF!
    they used the chain rule for the derivative
    You start out with ##\int F(x)dx## and make the substiution ##x =\frac{z}{1-z}## and obtain ##\int F(\text{substitute here})dx## you also need to substitute dx for (some stuff)*dz. That (some stuff) is the differential of the substitution, or in other words ##\frac{dx}{dz}##
  4. Apr 7, 2015 #3


    Staff: Mentor

    I'm positive that what you wrote is not what you meant.
    You have ##z = \frac x 1 + x## and ##x = \frac z 1 - z##.

    Since that's not what you intended, use parentheses around the denominators, like so:
    z = x/(1 + x), and x = z/(1 - z).
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