+ c_{\text{V}}=\frac{3\hbar c}{2\pi^2}\int_0^{\Theta_D/T}
+ \frac{x^3e^xx}{T(\beta\hbar c)^3(e^x-1)^2}\frac{dx}{\beta\hbar c}=
+ \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}\int_0^{\Theta_D/T}
+ \frac{x^4e^x}{(e^x-1)^2}dx
+ \]
+ \[
+ \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}=
+ \frac{3k_{\text{B}}}{2\pi^2\beta^3\hbar^3 c^3}=
+ \frac{3k_{\text{B}}T^33n}{\Theta_{\text{D}}^3}
+ \]
+ \[
+ \Rightarrow
+ c_{\text{V}}=9nk_{\text{B}}\left(\frac{T}{\Theta_{\text{D}}}
+ \right)^3\int_0^{\Theta_D/T}\frac{x^4e^x}{(e^x-1)^2}dx