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I have to find the state function of a gas (which under very low pressures behaves as an ideal gas). So, first I tried writing it as part of an equation $ dV = \dfrac{\partial V}{\partial T}\ dT + \dfrac{\partial V}{\partial p}\ dp $ and then checking if the "switched" derivatives will be equal. And then I tried to find an integrating factor as a function of $p$ and then $T$. The result in both cases was overly complicated which suggests that it is not the correct answer.

Could you please explain to me how to solve it? And when is it alright to change $V, p, T$ using the ideal gas equation to make calculations easier?

Moreover, I have a conceptual difficulty - what would the integrating factor times $dV$ represent? I only used integrating factor to prove that $1/T$ was the integrating factor of heat (which gives entropy).

the partial derivatives

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