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The isothermal equation of state $P=(c^{\rm{iso}}_{s})^2 \rho$ gives us that $c^{\rm{iso}}_{s} = \sqrt{RT}$. So we can write  \begin{equation}  \int{\frac{dP}{\rho}} = \frac{(c^{\rm{ad}}_{s})^2 }{\gamma -1} = \frac{\gamma (c^{\rm{iso}}_{s})^2 }{\gamma -1} \quad \rm{Adiabatic} \ \rm{Flow}  \end{equation}  \begin{equation}  \int{\frac{dP}{\rho}} = (c^{\rm{iso}}_{s})^2 \ln{\rho} \quad \rm{Isothermal} \ \rm{Flow}  \end{equation}