Fundamental Theorem of Calculus - Proof of The Corollary

Proof of The Corollary

Suppose F is an antiderivative of f, with f continuous on . Let

.

By the first part of the theorem, we know G is also an antiderivative of f. It follows by the mean value theorem that there is a number c such that G(x) = F(x) + c, for all x in . Letting x = a, we have

which means c = − F(a). In other words G(x) = F(x) − F(a), and so