$\int dx/x =\cdots$ (pedantic nitpicking?) It seems that "everybody knows" that $\displaystyle\int\frac{dx}{x}=\log|x|+C$ (or if one really must, then $\ln$ instead of they synonymous $\log$).  Does any textbook or reference work say that
$$
\int\frac{dx}{x} = \log |x| + \left.\begin{cases} A & \text{if }x>0, \\ B & \text{if }x<0. \end{cases}\right\} \text{where $A$ and $B$ are constant ?}
$$
And now a subtler question: When and how often would this matter?  Maybe never?  Or maybe only on examinations (Maybe there's some subtle difference between that and "never".)?
 A: It is typically understood that $\int f(x) dx$ denotes the set of antiderivatives of $f$ on an interval where $f$ has an antiderivative. 
The formula $\int \frac{dx}{x} = \ln |x|+ C$ is true over any interval $I \subset \mathbb R$ where $f$ is defined.
I am not an expert, but this could come in play when solving differential equations.
A: Actually this is a good point, there can be two different constants for the positive and negative log. You can get into trouble in some situations if you dont take this into account. It really is glossed over in all books. But these situations are quite rare. All of this stuff about constants of integration is not pedantic, it really affects the mathematics.  
A: I would say that even the constant of integration is not important and is more of a convention than anything else. Here is a quote from Spivak's Calculus (p. 361):

Most people write $$\int\! x^3 dx = \frac{x^4}{4} + C$$ to emphasize that the primitives of $f(x) = x^4$ are precisely the functions of the form $F(x) = x^4/4 + C$ for some number $C$. Although it is possible to obtain contradictions if this point is disregarded, in practice such difficulties do not arise, and concern for this constant is merely an annoyance.

Spivak continues to write the rest of the chapter without the constant of integration.
From a pedagogical perspective I think there is an advantage to requiring the constant of integration (and possibly even multiple constants of integration) but I don't think that it is very important as long as the user understands what they are doing.
If you interpret $\int f \,dx$ as asking for every function so that the derivative is equal to $f$ then you must use the multiple constants approach. If you interpret it as asking for some function then you can leave out any constant of integration. If you interpret it as every function on some interval where the antiderivative exists then the single constant approach is sufficient.
As long as everyone understands that every antiderivative over an interval differs from any other antiderivative by a constant then the constant of integration is not essential. Similarly for antiderivatives over disjoint unions and the multiple constants of integration approach.
