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.