The antiderivative of 1/x is ln(x), not ln |x

The antiderivative of 1x is commonly shown to be ln⁡|x|, since the graph of ln⁡x doesn't show anything where x<0. However, looking more closely, ln⁡|x| can be rewritten as follows:

ln⁡|x|={ln⁡(−x)if x<0ln⁡xif x>0

Additionally, since an antiderivative can be offset by an arbitrary constant, ln⁡|x|+3 is also a valid antiderivative. But going back to the piecewise function — since 1x is undefined at x=0, each piece can be offset by a different constant, like this:

∫1xdx={ln⁡(−x)+5if x<0ln⁡x+3if x>0

Using this fact, I could choose the constants in a specific way:

∫1xdx={ln⁡(−x)+iπif x<0ln⁡xif x>0

This is just as valid as an antiderivative, and is equivalent to taking the principal branch of ln⁡x for negative values of x.

In fact, if the domain is extended to complex numbers, ln⁡|x| is not differentiable at all in complex analysis, leaving only ln⁡x as the antiderivative.