Deriving Planck's Law: Solving for the Integral Formula

[raul_l]

Homework Statement

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

Homework Equations

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.

 

[Hootenanny]

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

Assuming that the formula above contains a typo and that the x^x was meant to be x³, that would mean you would like to evaluate

$$I = \int^{\infty}_{0} {\frac{x^3}{e^x-1}}dx}$$

In which case try http://en.wikipedia.org/wiki/Planck's_law_of_black_body_radiation#Derivation

 

[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. :)

Thanks for your help.