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 symmetrising factors.

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.