I am trying to derive a variable transformation in Frenkel & Smit Understanding Molecular Simulation: From algorithm to applications, on page 119. They are going over Monte Carlo integration in the NPT ensemble (constant N, P, T) where n is particles, P is pressure, T is temperature.
They convert the partition function to integrate phase space with respect to the logarithm of V rather than V itself(V is volume). I thought this would just be a simple change of variables but I can't reproduce it.
\begin{equation} \int dV V^n \exp(-\beta PV) \rightarrow \int d(\ln V) V^{n+1} \exp(-\beta PV) \end{equation}
Bounds are all phase space... i.e., negative infinity to infinity. Note that in the logarithm case, the factor $V^n$ is now $V^{n+1}$. $\beta = 1/T$. $\beta$, and P are considered constants.