Are all antiderivatives of a function an area or a (difference of areas) function of its derivative?

Question

If you take $f(t)=2t$ then the integral $\int_a^xf(t)dt=x^2-a^2$ seems to be the general antiderative of $f$ and the notation of $\int f(t)dt$ makes sense as it is saying the antideritave of $f$ is really some kind of general, varying definite integral. But $x^2-a^2$ isn't the general antiderivative of $f$ because the arbitrary constant is never positive. So for particular antiderivatives like $x^2+3$ it isn't any definite integral of $2t$, so the notation $\int f(t)dt$ doesn't seem to have any meaning. Am I simply wrong in saying that $x^2+3$ isn't any kind of definite integral or is notation sometimes just notation?