Throughout this handout, assume that *f*
is a continuous function on [*a*, *b*].

**FTC1
FTC2
(Proof)**

[rev. 11/8/2002]

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

(Q.E.D.)

**FTC2
FTC1
(Proof)**

[rev. 11/8/2002]

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

(Q.E.D.)