$$U=3\int \; <!--nolimits-->d\omega D(\omega )\frac{\hslash \omega }{exp<!--FUNCTION\; APPLICATION-->\left(\frac{\hslash \omega }{kT}\right)-1}=\frac{\hslash {\omega }_0}{exp<!--FUNCTION\; APPLICATION-->\left(\frac{\hslash {\omega }_0}{kT}\right)-1}3N.$$

$$C_V=\frac{dU}{dT}=\frac{3N\hslash \omega }{{\left(e^{\frac{\hslash \omega }{kT}}-1\right)}^2}\frac{e^{\frac{\hslash \omega }{kT}}\hslash \omega }{kT}.$$

$$C_V=3Nk.$$