Is it invalid to ask what the following function equals when $x=u$: $F(x)=\int_0^xf(u)(x-u)du$ I am new to integrals (only single variable) and am trying to parse how exactly to interpret the following function: $F(x)=\displaystyle\int_0^xf(u)(x-u)du$.
Expanding the interior symbols gives us: $\displaystyle F(x)=\int_0^x\left[f(u)x-f(u)u \right]du$. Presumably, we treat $x$ as a constant and conclude $F(x)=\displaystyle x \int_0^x f(u)\ du-\int_0^xf(u)u\ du$.
This is fine and dandy. My question is as follows:

Is it invalid (nonsense) to ask: "What does $F$ equal when $x=u$?"

In the context of this particular integral, $u$ is not a constant (rather, it is a function). Therefore, because $x$ is a constant, surely it makes no sense to state $x=u$. Is this correct?
If so, then it seems like $F(x)$ can be equivalently thought of as $\displaystyle \int_0^xf(u)\cdot g_x(u)du$ where $g_x(u)=x-u$, yes?
 A: The variable of integration - $u$ in this case - is what we call a "bound variable", meaning that it doesn't exist outside of the integral itself. Its sole purpose is to give us something to manipulate within the integral, but you aren't allowed to assign it a value since its values are already determined by context.
So the answer is yes - it is invalid to ask "what happens to $F(x)$ when $x = u$", although it is valid to talk about what happens within the integral at the point where $u = x$ (to which the answer is it's an endpoint of the integral which usually doesn't matter much).
A: I'm going to be very pedantic in this answer, because I think that for questions like this, it's valuable to carefully parse the exact meaning of the question
So, being very precise and only looking at the question exactly as it appears, the answer to the question

"What does $F$ equal when $x = u$?"

is simply "$F$".
The object that we are asked the value of is the function $F$, which in mathematical terms is an object in its own right, and so it is completely valid (though pointless) to make statements like $F = F$.
Moreover, the values that you do give information about ($x$ and $u$) do not appear in the object that you're asking about (the function $F$) and thus are totally irrelevant.

In fact, the careful answer to the seemingly more innocuous question

"What does $F$ equal when $x = 1$?"

is also "$F$" for the same reason.
That is, $x$ doesn't appear on the left-hand side, and so information about $x$ is irrelevant.
Of course, since you've previously defined $F$ in terms of $x$, a reader would most likely assume that the author's intention was to ask

"What does $F(x)$ equal when $x = 1$?"

Now, $x$ does appear in the object of interest (the element $F(x)$ of the codomain of $F$), so information about $x$ is relevant.

So having said this, there should be no reason to think that information about the value of $u$ gives any information about $F(x)$, since $u$ doesn't appear anywhere in the object that we're interested in!
The only reason that this raises confusion is that we have used the character $u$ when defining $F(x)$.
However, $u$ appearing in a variable of integration means that $F(x)$ is defined in terms of the values $f(u) (x - u)$ for all $u$ between $0$ and $1$, not just at a single point $u$.
Hence, we can rename the variable of integration $u$ without changing the value of $F(x)$ at all, so we shouldn't expect information about $u$ to have any relevance for the value $F(x)$.
