Newton thought of the function $F(x) = F(a) + \int_a^x f$
as solving the differential equation $y(a)=F(a), y'(x)=f(x)$
and thus being uniquely determined
(the above is poorly written/elaborated -- FIXME).

Newton was into solving differential equations, i.e.,
finding "fluents" for given "fluxions"
(integral curves for given vector fields,
 or at least anti-derivatives for given functions;
 this makes sense from the point of view of physics of course)