For an arbitrary antidifferentiable function $f(x)$, my goal is to construct a definite integral of $f(x)$: $$ \int_a^x f(t) dt $$
which is equal to one of the infinitely-many antiderivatives of $f(x)$, $F(x)+C$. Specifically, the one with a constant of integration $C=0$.
I've come up with the following relation between $a$ and $C$ which I think implies the existence of such an $a$: $$ \int f(x) dx = F(x) + C $$
$$ \int_a^x f(t) dt = F(x) - F(a) $$
$$ F(x) - F(a) = F(x) + C $$
$$ F(a) = -C $$
But, I don't know where to go from here.