1. Integrating factor of a 1-form

Giving a Frobenius integrable 1-form $\omega$ necessarily there exist a smooth function $F$ and a non vanishing smooth function $\mu$ such that

En tal caso, $\mu$ recibe el nombre de integrating factor y $F$ es una first integral of the Pfaffian system $\mathcal{S}(\omega )$. Tiene relación inversa con las symmetrizing factors.

For polynomial vector fields, the Darboux method constructs integrating factors from invariant algebraic curves via cofactor relations.

2. Integrating factor of an ODE

Given an ODE $\mathrm{\Delta }$, we call integrating factor of $\mathrm{\Delta }$ to any non vanishing smooth function $\mu \in J^m$ (the jet bundle) such that

for a smooth function $F$ called first integral.

To see the relation between integrating factor of an ODE and the integrating factor of the associated 1-form go to the note first integral#3 b First integal of an m th-order ODE.