finding the free energy of a van der waals gas (integration)

Question

I have the following integral,

$\int{ \frac{-nrtV}{(v-nb)^{2}} dV}$

could anyone tell me how to do this?

Answer 1

If $u = V - nb$, then $du = dV$, so

\begin{align}\int \frac{-nrtV}{(V - nb)^2}\, dV &= \int \frac{-nrt(u + nb)}{u^2}\, du\\ & = -nrt \int \left(\frac{1}{u} + \frac{nb}{u^2}\right)\, du \\ &= -nrt\left(\log|u| -\frac{nb}{u}\right) + C\\ &= -nrt\left(\log|V - nb| - \frac{nb}{V - nb}\right) + C. \end{align}