The meaning of the indefinite integral symbol the definition of an antiderivative 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?
 A: When we write$$\int f(x)\,\mathrm dx=F(x),$$what that means is that $F$ is an antiderivative of $f$, that is, it is a function with the same domain as $f$ such that $F'=f$. If it turns out that the domain of $f$ is an interval (with more than one point) then every antiderivative of $f$ will be equal to $F+K$, for some constant $K$.
And, indeed, if you have $f\colon(0,1)\longrightarrow\Bbb R$ defined by $f(x)=0$, then no constant function $F\colon\Bbb R\longrightarrow\Bbb R$ is an antiderivative of $f$, since they have distinct domains. However, the restriction of $F$ to $(0,1)$ is an antiderivative of $f$.
A: The statement $$\int f(x)\,\mathrm dx=F(x)$$ is a really interesting one. In my opinion, it's a good example of context-dependent notation: it doesn't mean the same type of thing in different scenarios.
Let's use a simple example for better intuition: $$\int 2x\,\mathrm dx=x^2+c$$
What is the right-hand side? Is it an expression? A family of expressions? Well, let's say my original question was:

Find $\int 2x\,\mathrm dx\ \ \ \ $ (1 mark)

Here, the meaning of the question is "find the functions ($\mathbb{R}\rightarrow \mathbb{R}$) that have derivative $2x$". And I've found them all: it's the set $\{f\in \mathbb{R}^{\mathbb{R}}:\exists c\in\mathbb{R}:\forall x\in\mathbb{R}:f(x)=x^2+c\}$ i.e. the set of functions $f:\mathbb{R}\rightarrow\mathbb{R}$ of the form $f(x)=x^2+c$ (for some $c\in\mathbb{R}$).
So $x^2+c$ is not an expression, but a set of functions. And so (for this to be "equal to" the left-hand side) the indefinite integral is a set of functions.
But what if my original question was:

Given that $\frac{\mathrm dy}{\mathrm dx}=2x$ and $y=f(x)$ and $f(0)=3$, find $f$.$\ \ \ \ \ \ $ (2 marks)

Now, when I write, $$f(x)=\int 2x\,\mathrm dx=x^2+c$$ it is the first step in a two-step solution (the second step is $0^2+c=3\implies c=3\implies f(x)=x^2+3$).
Here, the first step is the statement "for some specific and knowable value of $c$, and all real values $x$, the function $f$ satisfies $f(x)=x^2+c$".
It's not a statement about a set of functions, because I'm looking for one specific function. Suddenly, the indefinite integral is a single function: a specific antiderivative of $2x$ called $f$.
The indefinite integral sign, and surrounding presentation of calculations in calculus, is an example of abuse of notation: either there is no formal definition of a symbol, or the symbol is commonly used to indicate a very clear meaning, but one that violates the definition.
Moreover, functions are a common example of abuse of definitions: something isn't a function if it doesn't have both a domain and a codomain. So "$f(x)=x^2$ and $x>0$", which you might commonly see as a definition of the "function" $f$, is not a function, because I don't know whether the codomain of $f$ is $f(\mathbb{R}^+)=\mathbb{R}^+$, or $\mathbb{R}$, or even $\mathbb{C}$. Sometimes you don't even see the possible input values specified. And sometimes you see authors use definitions of functions where the image is irrelevant, and two functions are equal iff they have the same domain and the same value on every input in the domain.
Hopefully, the reason for all of this is clear: mathematical notation, like all forms of language, is primarily designed for easy communication. If there is no (reasonable) way to misinterpret the bunch of symbols you write on the page, then you are using them correctly.
A: 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.
