In calculus, it is said that
$$
\int f(x)\; dx=F(x)\quad\text{means}\quad F'(x)=f(x)\tag{1}
$$
where $F$ is a differentiable function on some open integral $I$. But the mean value theorem implies that any differentiable function $G:I\to \mathbb{R}$ with the property $G'(x)=f(x)$ on $I$ can be determined only up to a constant. Since the object on the right of the first equality of (1) is not unique, we cannot use (1) as a definition for the symbol $\int f(x)\;dx$.
Formulas for antiderivatives are usually written in the form of $\displaystyle \int f(x)\;dx=F(x)+C$. For example,
$$
\int \cos x\;dx = \sin x+C\;\tag{2}
$$
where $C$ is some "arbitrary" constant.
One cannot define an object with an "arbitrary" constant. It is OK to think about (2) as a set identity:
$$
\int \cos x\; dx = \{g:\mathbb{R}\to\mathbb{R}\mid g(x)=\sin x+C,\; C\in\mathbb{R}\}. \tag{3}
$$
So sometimes, people say that $\int f(x)\;dx$ really means a family of functions. But interpreting it this way, one runs into trouble of writing something like
$$
\int (2x+\cos x) \; dx = \int 2x\;dx+\int \cos x\; dx = \{x^2+\sin x+C:C\in\mathbb{R}\}\;\tag{4}
$$
where one is basically doing the addition of two sets in the middle, which is not defined.
So how should one understand the "indefinite integral" notation $\int f(x)\;dx$? In particular, what kind of mathematical objects is that?
Best Answer
This is precisely what is done.
When you move on to studying measure theory and consider $L^p$ spaces, two functions are considered "equal" if they only differ on a "small" set of points (where "small" has a precise measure-theoretic definition). Mathematicians are not computers, and know how to use the context of a statement to understand what version of equals is being used.
In the world of computing anti-derivatives, "=" means "differ by a constant", or more generally, "differ only by a constant on each connected component of their domains".
You can get into problems when you forget which version of "=" is intended, and think "=" means more than it does. (There are a few math brain-teasers out there based on that.) I think of it as the same problem as if you went into the teacher's lounge and asked for "the calculus teacher", as you were expecting Professor Liang, who is 6'4" tall and you wanted help getting something off a high shelf, but you didn't realize that Professor Smith, who is 4'11", also teaches calculus, and that's who shows up. You thought that specifying "calculus teacher" carried with it Prof. Liang's height, but that's not the case.