For inviscid, compressible, but isentropic ($s=$ constant) flow of a diatomic gas, the relationship between the pressure and density is $$p = A(s)\rho^{\gamma}\quad\text{where}\quad \gamma = \frac{c_p}{c_v} = \frac{7}{5}$$ Give Bernoulli's equation for such a gas flow. Explain why the pressure term is more important here than in a constant-density liquid flow.
Trying to search through the online notes there is section called Bernoulli's Equation which says: $$\frac{\partial\underline{u}}{\partial t} - \underline{u}\times\underline{\omega} = -\underline{\nabla}\mathcal{H}\quad\text{where}\quad \mathcal{H}(\underline{r},t) = \underline{\nabla}\bigg[\int\frac{1}{\rho}\mathrm{d}p + \frac{1}{2}||\underline{u}||^2 +\chi\bigg]$$ is the Bernoulli function, where assuming $p = p(\rho)$ we have $$\frac{1}{\rho}\underline{\nabla}p = \underline{\nabla}\bigg[\int\frac{1}{\rho}\mathrm{d}p\bigg]$$ Do I just rearrange so that: $$\rho = \exp\Bigg({\frac{5\ln\bigg(\frac{p}{A(s)}\bigg)}{7}}\Bigg) = \bigg(\frac{p}{A(s)}\bigg)^{5/7}$$ And then try to just sub in $\rho$ into the LHS or RHS?