What conventions surround the meaning of expressions like $\int \frac{1}{x} dx$?

I really struggle with the notation $\int f(x) dx$ because of the whole $+\,C$ thing, and this becomes double pronounced when $f(x)$ isn't defined everywhere. For example, we learned in high school that:

$$\int \frac{1}{x} dx = \log |x|+C$$

This doesn't really make sense to me. Personally, I think the correct answer is $$\int \frac{1}{x}dx = \log |x|+A \cdot H(x)+B\cdot H(-x),$$ where $H$ is the Heaviside step function.

Anyway, I want to understand the conventions surrounding this notation.

Questions.

What conventions surround the meaning of expressions like $\int \frac{1}{x} dx$? In particular:

Q0. How do mathematics educators typically understand the meaning of the notation $\int \frac{1}{x}dx,$ and would they give the solution $\log|x|+C$ full marks?

Q1. How do actual mathematicians understand the meaning of this notation? Is it simply avoided in serious mathematics? If not, would the statement $\int \frac{1}{x}dx = \log |x|+C$ be considered "correct"?

• Regarding the second inequality, it often is the case, by convention, in introductory Calculus that the functions are defined in an interval, making the equality equivalent to the first one. Commented Sep 8, 2015 at 10:07
• Dear @goblin : the tone is rather combative, and you are, by your own words, seeking opinions about what a large community thinks, which is not actually on-topic. Commented Sep 8, 2015 at 10:10
• @rschwieb, where is my tone combative? Commented Sep 8, 2015 at 10:10
• "I don't like conventional idea X. I think it should by Y. Anyway what does everyone think about this?" Not super combative, just a little. The more important thing here though is that it's discussion based rather than answerable. Commented Sep 8, 2015 at 10:12
• Regarding Q0, the problem is interpreting the symbol as integration of an expression instead of integration of a function (which will have a domain, hopefully an interval). In my opinion, though established, the notation is terribly unfortunate Commented Sep 8, 2015 at 10:12

I try to avoid writing $$\int \frac{1}{x}\,dx=\log|x|+C$$ to my students, since there is a common misunderstanding then, that, applying the fundamental theorem of calculus, $$\int_{-1}^2\frac{1}{x}\,dx = \log|2|-\log|-1|=\log 2,$$ even though the integral does not exist. Instead, I prefer to write $$\int\frac{1}{x}\,dx = \log x+C,\quad x>0,$$ and just mention that one has to be observant regarding which $x$'s one consider. Sometimes I also say that $$\int\frac{1}{x}\,dx=\log(-x)+C,\quad x<0,$$
The authors of the book we use write $$\int \frac{1}{x}\,dx=\log|x|+C$$ and therefore it would be strange not to give full credits on exams for that, if that question arises (I have not met this problem yet). I think that gives my point of view on your questions.
• I think that this is an important point. If you're trying to integrate $1/x$ past $0$, then you're probably doing something wrong. Therefore, we don't need different values of $C$ on different sides of $0$. Commented Sep 8, 2015 at 10:09
• Thanks for the thoughtful answer. Just a small critique, and I don't mean to pick a fight, but as a Year 12 student I always hated the notation $\mathrm{Fact}, \mathrm{Condition}$ because my brain reads this as $\mathrm{Fact},\mathrm{Fact}$. For instance, my brain reads $\int\frac{1}{x}\,dx = \log x+C, \; x>0$ as saying that $\int\frac{1}{x}\,dx = \log x+C$ is true, and that $x>0$ is also true. I think mathematics educators and professionals should choose a different notation for this, like: $$\int\frac{1}{x}\,dx=\log(-x)+C \;\;\bigg|\;\; x>0$$ where the chosen symbol is to be read "given." Commented Sep 8, 2015 at 20:01
• Or even just use a word: $$\int\frac{1}{x}\,dx=\log(-x)+C \;\;\mbox{if}\;\; x>0$$ Commented Sep 8, 2015 at 20:03
• I see, and maybe that way of writing it is a bit sloppy. Sometimes, you see $\forall x>0$ instead of just $x>0$. Commented Sep 8, 2015 at 20:04