Why does this device for solving for a function given a particular partial derivative work?

Question

In many problems in physics (in elementary thermodynamics in particular), one has an equation for a particular partial derivative of some other (potential) function. For example, if the "thermodynamic variables" in my (simple) system are $S,V,N$, I might have the partial derivative of $P$ with respect to $V$ and let's say it's constant for simplicity, $C$. Then what one usually sees is something like "integrating" this partial derivative to find $$P = \int \left(\frac{\partial P}{\partial V} \right)_{S,N}dV = \int C \, dV = CV + f(S,N)$$ where $f$ is some "constant" with respect to $V$. Is all this simply a device for noticing that the RHS of the above is an antiderivative (with respect to the partial with respect to $V$) of $\left(\frac{\partial P}{\partial V} \right)_{S,N}$, and thus (by a theorem seen in any elementary calculus text about two functions which have the same derivative; see e.g. Corollary 2 to Theorem 11-4 in Spivak's Calculus) differs by at most a constant (which we can just tacitly take inside of $f$ at any rate) from $P$ itself?