Suppose it is known that a given function ƒ( *x*) is the derivative of some function ƒ( *x*); how is ƒ( *x*) found? The answer, of course, is to integrate ƒ( *x*). Now consider a related question: Suppose it is known that a given function ƒ( *x, y*) is the *partial* derivative with respect to *x* of some function *F*( *x, y*); how is *F*( *x, y*) found? The answer is to integrate ƒ( *x, y*) with respect to *x*, a process I refer to as **partial integration**. Similarly, suppose it is known that a given function ƒ( *x, y*) is the partial derivative with respect to *y* of some function ƒ( *x, y*); how is ƒ( *x, y*) found? Integrate ƒ( *x, y*) with respect to *y*.

**Example 1**: Let *M*( *x, y*) = 2 *xy* ^{2} + *x* ^{2} − *y*. It is known that *M* equals ƒ _{x }for some function ƒ( *x, y*). Determine the most general such function ƒ( *x, y*).

Since *M*( *x, y*) is the partial derivative with respect to *x* of some function ƒ( *x, y*), *M* must be partially integrated with respect to *x* to recover ƒ. This situation can be symbolized as follows:

Therefore,

Note carefully that the “constant” of integration here is any (differentiable) function of *y*—denoted by ψ( *y*)—since any such function would vanish upon partial differentiation with respect to *x* (just as any pure constant *c* would vanish upon ordinary differentiation). If the question had asked merely for *a* function ƒ( *x, y*) for which ƒ _{x }= *M*, you could just take ψ( *y*) ≡ 0.

**Example 2**: Let *N*( *x, y*) = sin *x* cos *y* − *xy* + 1. It is known that *N* equals ƒ _{y }for some function ƒ( *x, y*). Determine the most general such function ƒ( *x, y*).

Since *N*( *x, y*) is the partial derivative with respect to *y* of some function ƒ( *x, y*), *N* must be partially integrated with respect to *y* to recover ƒ. This situation can be symbolized as follows:

Therefore,

Note carefully that the “constant” of integration here is any (differentiable) function of *x*—denoted by ξ( *x*)—since any such function would vanish upon partial differentiation with respect to *y*. If the question had asked merely for *a* function ƒ( *x, y*) for which ƒ _{y }= *N*, you could just take ξ( *x*) ≡ 0.