 Solution

The harmonic phonon contribution to the internal energy isFor $T\gg {\Theta }_D$, every retained mode has $x=\hslash \omega /(k_{B}T)\ll 1$. Expanding $1/(e^x-1)=1/x-1/2+O(x)$ cancels the explicit half quantum and gives $k_{B}T$ per mode to leading order. ThereforeEquivalently, differentiating before integrating givesThe integrand tends to $x^2$, and its integral is $({\Theta }_D/T{)}^3/3$, producing the same Dulong-Petit law. In classical language, the equipartition theorem assigns $k_{B}T/2$ each to the kinetic and elastic parts of every harmonic normal mode. Fixed zero-point energy does not contribute to the heat capacity.