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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.

Question

Its answer to part (b) is also below.

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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.

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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.

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