Stefan-Boltzman's radiation law problem

[steveurkell]

Hello everyone,
I cannot solve the following integral
integral from 0 to infinity of (x^3)/(exp(kx)-1) dx with k>0
as you may be able to guess, this integral is very famous one which can be used to deduce Stefan-Boltzman's radiation law
those of you who are experts may quickly recognize it and easily solve it
I've ever learned the trick 4 years ago, now forget hehe
so I would be grateful if you could teach me that trick
any comments are welcome

regards

 

[dextercioby]

1.
HINT:Write the integrand as a product between the
$$x^{3}$$
and the fraction containing the exponetial;then write the fraction as a sum of a geometic series with ratio $e^{-kx}$.
Take the sum in the exterior of the integral and then perform three times partial integration.U'll end up with:
$$I=6\sum_{k=1}^{+\infty} \frac{1}{k^{4}}$$

which is
$$I=6\zeta(4)$$

2.Search the integral in the tables (e.g.Abramowitz & Stegun,Gradsteyn & Rytzik ) to find
$$I=\Gamma(4)\zeta(4)$$

Daniel.

 

[steveurkell]

thank you
before my first posting, I had tried to use the geometrical series expansion, but I performed the summation inside the integral and I didn't realize that with limit of integration from 0 to infinity, I just could replace x by 0.

thanks anyway