If a function f has an antiderivative F, that means the derivative (slope) of F at x is f(x).
Now, suppose you add a constant C to F(x), giving a new function G(x) = F(x) + C. Then F and G have the same derivative (namely, f) so that G is also an antiderivative of f. It goes the other way, too. Suppose two functions F and G are antiderivatives of f. Then their difference H(x) = G(x) - F(x) has a derivative of zero since the derivatives of G and F are both f. So H has zero slope and so must be a constant C.
In summary, if you add a constant C to F, you get another antiderivative G (= F + C) of f, and if F and G are antiderivatives of f, then they differ by a constant C.
So suppose you have two antiderivatives F and G of f. Then there is a constant C such that G = F + C. Hence,
F(b) - F(a) = G(b) - C - [ G(a) - C ]
= G(b) - G(a)
