# Integrating a function from 0 to infinity correctly?

1. Apr 14, 2013

### ck99

1. The problem statement, all variables and given/known data

I am trying to integrate the PLanck function to get the Stefan Boltzmann law. After factoring out constants, and substituting x = hv/kT I am left with the following integral:

B(T) = ∫ x3/(ex - 1) dx integrated from 0 to ∞

The next step in my notes is that the result of this integral is ∏4/15 and I have no idea how this is done!

2. Relevant equations

3. The attempt at a solution

I tried Wolfram with this and it gave such a complicated answer that I was even more confused - certainly it looked nothing like the one in the notes! I know asking for help shouldn't be my first resort, but have no idea how to begin tackling this I'm afraid.

2. Apr 14, 2013

### alejandrito29

$$\int_0^{\infty} \frac{x^3 dx}{e^x-1} =\int_0^{\infty} \frac{x^{4-1} dx}{e^x-1} = \zeta (4)\Gamma (4) = \frac{\pi ^4}{90}\cdot 3! = \frac{\pi ^4}{15}$$

cosmology at Chile University? ???

3. Apr 15, 2013

### ck99

Ah, thanks for the help but I think that derivation is beyond the scope of this course. (certainly it's beyond me!)