Substitution rule

Substitution rule: Let $\alpha<\beta$, let $\varphi:[\alpha,\beta]\to\mathbb{R}$ be continuously differentiable and strictly increasing, and set $a=\varphi(\alpha)$ and $b=\varphi(\beta)$. If $f:[a,b]\to\mathbb{R}$ is continuous, then

This is the one-dimensional instance of the general change of variables formula, and it is closely tied to the chain rule and fundamental theorem of calculus II.