Integral of $\frac1x$ When integrating $\frac1x$, you would get $\ln|x|+c$. Working under that assumption, any given derivative of $\ln|ax|+c$ would give you the same answer as any derivative of $\ln|x|+c$.  Given this, after deriving and then integrating, $\ln|2x|+c$ and $\ln|100x|+c$ would have the same solutions of $\frac1x$ after integrating and $\ln|x|+c$, even though the two original equations have differing solutions, the resulting equation are identical. Is there a way to differentiate the equations even after deriving and integrating?  Are the solutions found after this process incorrect?

$$\ln|ax|+C = \ln|a| + \ln|x| + C = \ln|x| + C$$
Note that $\ln|a|$ is constant, so we can simply wrap it into $C$
