In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.

Consider an ordinary differential equation of the form

Consider a function Failed to parse (Missing texvc executable; please see math/README to configure.): M(x) . We multiply both sides of (1) by Failed to parse (Missing texvc executable; please see math/README to configure.): M(x):

We want the left hand side to be in the form of the derivative of a product (see product rule), such that (2) can be written as

The left hand side in (3) can now be integrated