+ c_{\text{V}}=\frac{\partial}{\partial T}\, {\color{red}3}\int_0^{k_D}
+ {\color{blue}\frac{k^2}{2\pi^2}dk} \frac{\hbar {\color{green}ck}}
+ {e^{\beta\hbar {\color{green}ck}}-1}=
+ \frac{\partial}{\partial T}\frac{3\hbar c}{2\pi^2}\int_0^{k_D}
+ \frac{k^3}{e^{\beta\hbar ck}-1}dk=
+ \frac{3\hbar c}{2\pi^2}\int_0^{k_D}
+ \frac{k^3e^{\beta\hbar ck}\beta\hbar ck\frac{1}{T}}
+ {(e^{\beta\hbar ck}-1)^2}dk
+ \]
+ Change of variables: $\beta\hbar ck=x$
+ \[
+ \Rightarrow
+ k=\frac{x}{\beta\hbar c} \quad \textrm{, } \quad
+ dk=\frac{1}{\beta\hbar c} dx
+ \]
+ \[
+ c_{\text{V}}=