Solving Change of Variables: x^3 / (e^x - 1) dx

[Ryan000]

Homework Statement

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).

Homework Equations

integral of f(x) dx

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!

 

[BiGyElLoWhAt]

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 $\frac{dx}{dz}$

 

[Mark44]

Homework Statement

In the book it gave the example for standard change of variables as,
z = x / 1 + x or equivalently x = z / 1 - z , then

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).

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).

Homework Equations

integral of f(x) dx

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!