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"?
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.