Integration problem to calculate partition function of a gase in a blackbody

[PullMeOut]

Homework Statement

This is the integration i have to solve
I=$$\int x^{2}In(1-exp(-ax))dx$$
integration is from zero to infinity

The Attempt at a Solution

I know that it should be solved with integration by parts
so
u=In(1-exp(-ax))
du=[a exp(-ax)] / [1-exp(-ax)]
dv=x$$^{2}$$dx
v=x$$^{3}$$ /3
when i put this into the integration formula
I=u*v-$$\int v*du$$
it becomes more complicated
I=In(1-exp(-ax))*x$$^{3}$$/3 - $$\int dx * (x^3/3) * [a exp(-ax)] / [1-exp(-ax)]$$
so what should i do after this , i can't figure it out, am i doing it wrong?

 

[tim_lou]

Use Gamma function and expand integrand using geometric series.

 

[PullMeOut]

if i do that i should calculate the series from zero to infinity. when will i know that i should stop?

 

[tim_lou]

You should get some good looking series at the end, and you can get a closed form expression. Usually, one ends up with some kind of zeta functions.

You might get things like:
$$\sum \frac{1}{n^2}=\frac{\pi^2}{6}$$

$$\sum \frac{1}{n^4}=\frac{\pi^4}{90}$$

which can be simplified by consulting some tables.

 

[PullMeOut]

well thank you for your help. now i will give it a try.