# When was the convention for the indefinite integral $\int\frac{1}{x}dx$ changed?

In Europe, in the 20th century, $\int\frac{1}{x}dx$ equalled $\ln{x}+C$. (I have references from Poland for 1930-1947 and the UK for the 1960s and 1970s).

Now, if one mentions $\int\frac{1}{x}dx=\ln{x}+C$ in Mathematics Stack Exchange, one is lynched. The doctrine is now that $\int\frac{1}{x}dx=\ln|x|+C$, and any suggestion to the contrary is a crime.

I don't want to discuss the pros and cons of this alteration, but I am interested to know: who made this change, and when? Was it intentional, or just an influential textbook? Or is this less of an "old millennium/new millennium" and more of an "Old World/New World" kind of thing?

There is an argument for posting this to Maths Stack Exchange rather than here, and I will if asked; but it is a historical question and so this seems a reasonable forum for it.

• The doctrine is $\ln|x| + C$? I thought it is $\ln|x| + C_1$ for $x > 0$ and $\ln|x| + C_2$ for $x < 0$, where $C_1$ and $C_2$ are constants that are not necessarily equal. – KCd Aug 24 '16 at 4:16
• This question is of the form "when did X happen?," where X never happened. What's being discussed here is not a "convention." Definitions could be described as conventions, but the value of $\int dx/x$ is not a definition, it's a consequence of a definition. The inflammatory language about lynching and crimes is not helpful in starting a reasoned discussion. – Ben Crowell Aug 24 '16 at 5:48
• @Kcd: shouldn't that be $\ln x + C_1$ for $x > 0$ and $\ln (-x) + C_2$ for $x < 0$? – Jonathan Cast Aug 24 '16 at 15:06
• This is not a "convention": the first formula is only true when $x>0$ (if we stay in the real domain. – Alexandre Eremenko Aug 24 '16 at 17:29
• About the "convention". If $\ln$ is taken as an analytic branch the first formula is correct. If one restricts $\ln$ to positive integers only, then one needs the absolute value to get a formula that is also valid for negatives. But what one gets then is not a restriction of any single branch of $\ln$ to the union of positives and negatives. The domain, the meaning of $\ln$, and the particular antiderivative in front of $C$ do not follow from definitions, and are conventional. Textbooks switching from $\ln x$ to $\ln|x|$ did happen, it was gradual rather than an event though. – Conifold Aug 24 '16 at 21:01

Mathematicians say $\int \frac{1}{x}\;dx = \log x + C$. It works even for the complex case.
Calculus instructors say $\int \frac{1}{x}\;dx = \ln|x|+C$ for some reason, but it is WRONG in the complex case. (Perhaps that "ln" in there gives us a hint that they are writing for engineers and physicists rather than mathematicians.)