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I recently stumbled across a claim that $\ln |x| + C$ isn't the most general antiderivative of $1/x$. The argument was that the parts of the curve $\ln |x|$ separated by the $y$-axis do not have to be shifted by the same constant $C$. Therefore, an even more general antiderivative would be:

$ \displaystyle \int \frac{1}{x}\mathrm{d} x =\begin{cases} \ln x+D & x\gt 0 \\ \ln(-x) + C & x \lt0 \\ \end{cases} $

Where $D, C \in\mathbb{R}$ are real constants which may or may not be the same.

I am wondering why we typically say that the antiderivative of $1/x$ is $\ln |x| + C$, when there seems to be an even more general function. Does it make any practical difference to even make a distinction between these functions?

J. W. Tanner
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Anna
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    See https://math.stackexchange.com/q/2720780/42969 and https://math.stackexchange.com/q/4466926/42969 – Martin R Jun 05 '24 at 11:44
  • The $+C$ in an indefinite integral is a locally constant function, so strictly speaking it has an invisible $(x)$. It matters here because the domain of $\frac1x$ is in two pieces. – J.G. Jun 05 '24 at 12:28

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