The indefinite integral $F(x) = \int f(x) dx$ is just a function whose derivative is $f(x)$. There are an infinite number of such functions, because for any constant C you can take $F_C(x) = F(x) + C$ and it's still a valid antiderivative.
So the particular value of $F(x)$ at any given $x$ is not actually that meaningful, unless we've done something to select a single function out of the infinite possibilities.
For example, if an object travels at constant velocity $v$, then the integral of its velocity with respect to time is $s(t) = \int v\ dt = vt + c$ for an arbitrary value $c$. On its own, this is not hugely useful, but there are two useful things we can do with it:
We can look at the difference of $s(t)$ between times $t_1$ and $t_2$, which gives us $s(t_2) - s(t_1) = (vt_2 + c) - (vt_1 + c) = v(t_2 - t_1)$, which is the distance traveled between the two time points.
Or, (and in this case it's actually roughly equivalent), we can say that we know that the object had displacement 0 at time $t = 0$, i.e. we fix $s(0) = 0$ which then tells us that $c = 0$, and so now we have a fixed function of displacement $s(t) = vt$.