Why is $\int \frac{f'(x)}{f(x)}=\ln |f(x)|$ ignored in differential equations? I've noticed that whenever $\int \frac{f'(x)}{f(x)}$ comes up in a differential equation the answer is always given as $\ln f(x)$ rather than $\ln |f(x)|$ as I was taught it should be. Is it because of the arbitrary constant? In other words, since $$\int \frac{f'(x)}{f(x)}=\ln |f(x)|+\ln A$$ for some constant $A$, then the answer is $\ln A|f(x)|$ and because $A$ can be positive or negative it follows that there is no point including the absolute signs? Hence the answer is given as $\ln f(x)+C$ for some constant $C$ rather than $\ln |f(x)|+C$. Is this why?

Writing $ \int \frac{f'(x)}{f(x)} = \ln f(x)$ implicitly supposes that $f(x) > 0$. The logarithm is not defined for negative numbers.

Why isn't the general formula needed?
Consider the differential equation
$$
y' = ty.
$$
To find the solutions to such equations, we can proceed in two steps.
Step 1. Suppose we have a solution $y: I \rightarrow \mathbb{R}$, where $I$ is an open interval. Further assume $y(t) \not = 0$, for all $t$. Since $I$ is an interval, then either $y > 0$ or $y < 0$. In the first case, we find $\frac{y'(t)}{y(t)} = t$, from which $\ln y(t) = t^2/2 + C$ and $y(t) = \alpha e^{t^2/2}$, for some $\alpha > 0$.
Step 2. We verify that $y(t) = \alpha e^{t^2/2}$, $I = \mathbb{R}$, is indeed a solution for all $\alpha \in \mathbb{R}$ and that all solutions have been found.
As you can see, when everything is done so carefully, there is no need to use the general formula $\int \frac{f'}{f} = \ln|f|$. I guess this is what teachers have in mind when presenting an abbreviated argument and writing "$\int \frac{y'(t)}{y(t)} = \ln y(t)$".
