Solution
= Solution
A <phonon> has zero <chemical potential> and energy $\hbar\omega$. The total <internal energy> is consequently
$$
E_{\rm tot}
=\int_0^{\omega_D}
\frac{\hbar\omega\,g(\omega)}{e^{\beta\hbar\omega}-1}\,d\omega.
$$
With the <change of variables formula>[change of variable] $x=\beta\hbar\omega$ and $\beta\hbar\omega_D=T_D/T$, this becomes
$$
\boxed{
E_{\rm tot}
=\frac{B(k_BT)^{b+2}}{\hbar^{b+1}}
\int_0^{T_D/T}\frac{x^{b+1}}{e^x-1}\,dx,}
\qquad b=\frac2\alpha-1.
$$