Deriving Planck's law

1. Nov 11, 2008

raul_l

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

I need to find the Planck's law: $$R(\lambda)=\frac{2hc^2}{\lambda^5}\frac{1}{e^{\frac{hc}{\lambda kT}}-1}$$

2. Relevant equations

3. The attempt at a solution

I've done most of the derivation, but I got stuck with an integral: $$R(\lambda)=\frac{1}{4\pi^3 \hbar^3 c^2} \int^{\infty}_{0} {\frac{E^3}{\exp{{\frac{E}{kT}}}-1}}dE}$$

Basically, I need a formula for $$\int^{\infty}_{0} {\frac{x^x}{e^x-1}}dx}$$

Could anyone give me the formula or perhaps a link where I could find it myself or maybe just point me in the right direction somehow?

Thank you.

2. Nov 11, 2008

Hootenanny

Staff Emeritus
3. Nov 11, 2008

raul_l

Found it: $$\int^{\infty}_{0} {\frac{x^3}{e^x-1}}dx}=\frac{\pi^4}{15}$$

And evidently I don't really have to use it. :)