In the solutions of many mathematical problems requiring differential equation solving, I came across to the absence of absolute value with $\ln(x)$.
I can explain better with an example.
Let's consider the question below.
Its answer to part (b) is also below.
In the first A1 mark, why there is no absolute value resulting the integration of $-\tan{x}$?
We saw this behaviour on many other occasions.
You are trying to find an integrating factor not all integrating factors.
Integrating factors are facilitators and we just need one of them to solve our problem.