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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"?

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    $\begingroup$ 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. $\endgroup$
    – Git Gud
    Commented Sep 8, 2015 at 10:07
  • $\begingroup$ 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. $\endgroup$
    – rschwieb
    Commented Sep 8, 2015 at 10:10
  • $\begingroup$ @rschwieb, where is my tone combative? $\endgroup$ Commented Sep 8, 2015 at 10:10
  • $\begingroup$ "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. $\endgroup$
    – rschwieb
    Commented Sep 8, 2015 at 10:12
  • $\begingroup$ 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 $\endgroup$
    – Git Gud
    Commented Sep 8, 2015 at 10:12

1 Answer 1

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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.

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  • $\begingroup$ 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 $0$. $\endgroup$
    – Arthur
    Commented Sep 8, 2015 at 10:09
  • $\begingroup$ 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." $\endgroup$ Commented Sep 8, 2015 at 20:01
  • $\begingroup$ Or even just use a word: $$\int\frac{1}{x}\,dx=\log(-x)+C \;\;\mbox{if}\;\; x>0$$ $\endgroup$ Commented Sep 8, 2015 at 20:03
  • $\begingroup$ I see, and maybe that way of writing it is a bit sloppy. Sometimes, you see $\forall x>0$ instead of just $x>0$. $\endgroup$
    – mickep
    Commented Sep 8, 2015 at 20:04

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