The integrating factor \(\rho(x)\) is a value that helps undo the product rule. For which:

\begin{equation} log(\rho(x)) = \int P(x)dx \end{equation}

for some function \(P(x)\).

Separating the \(\rho(x)\) out, we have therefore:

\begin{equation} e^{\int P dx} = \rho(x) \end{equation}

Why is this helpful and undoes the product rule? This is because of a very interesting property of how \(\rho(x)\) behaves.