Throughout this handout, assume that f is a continuous function on [a, b].
Assume that the FTC1 method of finding definite integrals is valid, i.e., , where G is any antiderivative of f.
Let d [a, b] be a constant, and let G
be the designated antiderivative. Then
Assume that the FTC2 conclusion is valid, i.e.,
the derivative of an “accumulator function” of x (where the accumulator is defined as a definite integral with
variable upper endpoint x) is merely
the integrand evaluated at x. Let d [a, b] be a constant, and
let H be the accumulator function where . By FTC2, therefore, H is an antiderivative
of f. Since any two antiderivatives differ by a constant, any antiderivative G can be written as G(x) = H(x) + C,
where C is a constant. Equivalently, H(x)
= G(x) − C. Then