Integrating Factor for First Order ODE

Theorem
Let the first order ordinary differential equation:
 * $(1): \quad M \left({x, y}\right) + N \left({x, y}\right) \dfrac {\mathrm d y} {\mathrm d x} = 0$

be non-homogeneous and not exact.

By Existence of Integrating Factor, if $(1)$ has a general solution, there exists an integrating factor $\mu \left({x, y}\right)$ such that:
 * $\displaystyle \mu \left({x, y}\right) \left({M \left({x, y}\right) + N \left({x, y}\right) \frac {\mathrm d y} {\mathrm d x} }\right) = 0$

is an exact differential equation.

Unfortunately, there is no systematic method of finding such a $\mu \left({x, y}\right)$ for all such equations $(1)$.

However, there are certain types of first order ODE for which an integrating factor can be found procedurally.

Proof
We have one of these:
 * Integrating Factor for First Order ODE: Function of One Variable: $x$ or $y$ only
 * Integrating Factor for First Order ODE: Function of $x + y$
 * Integrating Factor for First Order ODE: Function of $x y$