TITLE: The meaning of the indefinite integral symbol the definition of an antiderivative
QUESTION [5 upvotes]: I have been thinking that the symbol $$\int f(x) dx$$ is a variable as a whole, referring to $F(x)$, where $F$ is some antiderivative of $f$. It can refer to any function that is an antiderivative of $f$, evaluated at $x$. So $A =$ the symbol means $A$ is equal to some function evaluated at $x$, and that function is some unknown antiderivative of $f$.
Does this symbol mean above? Am I correct?
Also,
What is the precise definition of an antiderivative of a function? What is the definition that is most commonly accepted? What is the one that I write, others will know what I am referring to? Is there one?
I need to know it because the symbol is based on an “antiderivative”.
According to Wikipedia, an antiderivative of $f$ is $F$ such that $F' = f$. And two functions are equal iff their domains are the same and outputs are the same for each input from the domain.
By that definition, $F: \mathbb{R} \rightarrow \mathbb{R}$, $F(x) = C$ is not an antiderivative of $f: (0,1) \rightarrow \mathbb{R}$, $f(x) = 0.$
Is this also correct? Is it the commonly accepted definition of an antiderivative?

REPLY [4 votes]: To directly answer your question, $F$ is an antiderivative of $f$ if and only if $F' = f$. In this case, we can write $$F(x) = \int f(x) \ \text dx$$ This is the most common (and arguably, the only reasonable) definition of the word.
However, I think the source of confusion here is not your understanding of antiderivatives but that of functions. We like to throw around symbols and write things like $f = g$ because it's concise and precise, but it's not exactly clear without some prior understanding. When we define functions like
$$ f: X\to Y \\ f: x\mapsto y$$
we're saying a lot of different things. First, there are sets $X$ and $Y$, called the domain and codomain, respectively. Then, there is a rule that associates to each $x\in X$ a particular $y\in Y$ (usually written as an expression involving $x$). We can think of the function as a triple: (domain, codomain, rule). Finally, $f$ is simply the name given to that function.
The upshot of all this is that a statement like $f = g$ means that the whole triple of (domain, codomain, rule) is the same between $f$ and $g$. Importantly, if $f$ and $g$ have different domains, then they are not the same function, even if they agree everywhere their domains overlap. This is why the statement at the end of your question is correct; $F'(x) = f(x)$ for $x\in (0,1)$, but $F'$ and $f$ have different domains and so are different functions.