Z, <U>, and C for Hagedorn Spectrum

[dark_matter_is_neat]

Homework Statement

The Hagedorn Spectrum with E>=0 is $\frac{dn}{dE} = \alpha E^{3} e^{B_{0}E}$ where $B_{0} = 1/kT_{0}$ where $T_{0}$ is the Hagedorn Temperature. For $T<T_{0}$ determine the partition function Z, the average energy <U> and the specific heat C for the system.

Relevant Equations

$\frac{dn}{dE} = \alpha E^{3} e^{B_{0}E}$

So to get the partition function I do the integral $\int \alpha E^{3} e^{(B_{0}-B)E} dE$, which substituting in $/Delta B = B_{0} - B$ is $Z = \frac{ \alpha E^{3} e^{\Delta B E}}{\Delta B} - \frac{3 \alpha E^{2} e^{\Delta B}}{\Delta B^{2}} + \frac{6 \alpha E e^{\Delta B E}}{\Delta B ^{3}} - \frac{6 \alpha e^{\Delta B E}}{\Delta B ^{4}}$. To get <U> you just calculate $- \frac{\partial ln(Z)}{\partial B}$ and to get C you just calculate $k \frac{\partial <U>}{ \partial T}$. The math gets pretty rough calculating <U> and gets especially rough calculating C and I'm wondering if there is a much less tedious way to get them or if I messed up calculating Z.