Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 10D / a / Solution

Substitute $x=pc/(k_{B}T)$ in the Fermi-Dirac distribution energy integral. ThenThe integral converges, is positive, and is independent of temperature. Thus $ϵ=\alpha T^4$, with $\alpha$ equal to the displayed constant integral. The given normalization is for the specified degeneracy; further internal degrees of freedom multiply it.