积分因子是一种用来解微分方程的方法。

考虑以下形式的微分方程：

其中是的未知函数，和是给定的函数。

我们希望把左面化成两个函数的乘积的导数的形式。

考虑函数。我们把(1)的两边乘以

如果左面是两个函数的乘积的导数，那幺：

两边积分，得：

其中是一个常数。于是，

为了求出函数，我们把(3)的左面用乘法定则展开：

与(2)比较，可知满足以下微分方程：

两边除以，得：

等式(5)是对数导数的形式。解这个方程，得：

我们可以看到，的性质在解微分方程中是十分重要的。称为积分因子。

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.