Getting rid of absolute value in integrating factor

Question

If I have this equation $$|I|=e^C |x^3|$$ where $C$ is a constant, yet to be determined. Is it allowed to say:

let $A$ be a constant such that $$\begin{cases} A=-e^C \space\space\space if\space\space \frac{I}{x^3}<0 \\ A=e^C \space\space\space\space\space\space if\space\space \frac{I}{x^3}\gt0 \end{cases}$$

then $$I=Ae^x$$

This is part of an integration using the integrating factor method, btw.

Answer 1

Simple answer: yes. Since $C$ is arbitrary to begin with, $A$ will then be an arbitrary nonzero constant.

(But how do you come across this in the context of integrating factors? The place where it usually shows up all the time is integration of ODEs by separation of variables.)