I am trying to understand a solved exercise. It is about calculating the energy of vibrating bosons using the density of states, which is defined as follows:
$$g(\omega) = \frac{9N}{\omega^3_D} \omega$$
Afterwards, the specific heat $c_v$ is calculated ($E = \frac{\partial E}{\partial T}$).
After giving some context, I will go to the point.
$$E = \int_{0}^{\infty} \frac{\hbar \omega g(\omega)}{e^{\beta \hbar \omega} - 1} d\omega = \frac{9N}{\omega^3_D} \int_{0}^{\omega_D} \frac{\hbar \omega^3}{e^{\beta \hbar \omega} - 1} d\omega $$
On this integral, I have to use the following change of variables:
$$x = \beta \hbar \omega$$
$$T_D = \frac{\omega_D \hbar}{K_B}$$
After applying the first one I got:
$$E =\frac{9N}{\omega^3_D \beta^4 \hbar^3} \int_{0}^{\frac{\omega_D}{\beta \hbar}} \frac{ x^3}{e^{x} - 1} dx$$
Where:
$$\beta = \frac{1}{K_BT}$$
Once at this point I applied the second change of variables and derived E with respect to T (Temperature). My issue is that the solved problem skips this calculation and gives directly the $c_v$:
$$ c_v = 9K_BN (\frac{T}{T_D})^3 \int_{0}^{\frac{T_D}{T}} \frac{ x^4 e^x}{(e^{x} - 1)^2} dx$$
But I am not getting this outcome. May you help me out?