Integrating a function from 0 to infinity correctly?

[ck99]

Homework Statement

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/(e^x - 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!

Homework Equations

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.

 

[alejandrito29]

Homework Statement

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/(e^x - 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!

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.

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

cosmology at Chile University? ?

 

[ck99]

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