## 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:

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

*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:

But sincef(s(t)) s'(t) dt = F(s(b)) - F(s(b))

*F*is by assumption an antiderivative of

*f*we have that

which finishes the proof.