Suppose we have a piecewise-defined function $f$ such that $$x<0 \implies f(x) = \ln -x + c_1 \implies f'(x) = -\frac1{-x} = \frac1x$$ $$x>0 \implies f(x) = \ln x + c_2 \implies f'(x) = \frac1x$$
Then by the definition of antidifferentiation, $f(x)$ is a solution of $\int\frac1xdx$. However, in math class, what I have been told was that $\int \frac1x dx = \ln |x| +c$, with only one constant term. Is there in fact a constraint on the two constant terms such that they have to be equal, to result in the simplification to a single constant term, or is the one-constant solution just not general?
The constant terms on the two sides of a discontinuity can be different. This is clear by looking at the graph: you can shift up and down the graphs of the two sides independently without affecting the derivative.