Theorem 7.2.3: Substitution Rule

If f is a continuous function defined on [a, b], and s a continuously differentiable function from [c, d] into [a, b]. Then

Proof

Proof:

f is continuous so that there exists a function F with F' = f (in other words, F is an antiderivative of f). Differentiate the function F(s(x)) using the Chain Rule:

F(s(x)) = F'(s(x)) s'(x) = f(s(x)) s'(x)

because F' = f. Therefore the composite function F(s(x)) is an antiderivative of f(s(x)) s'(x) so that by our evaluation shortcut we have:

f(s(t)) s'(t) dt = F(s(b)) - F(s(b))

But since F is by assumption an antiderivative of f we have that

which finishes the proof.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007