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: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.