Integration/Substitution/dx version/Fact

Substitution rule II

Suppose that

$f\colon [a,b]\longrightarrow \mathbb {R}$

is a continuous function, and let

$\varphi \colon [c,d]\longrightarrow [a,b],s\longmapsto \varphi (s),$

denote a bijective continuously differentiable function.

Then

${}\int _{a}^{b}f(t)\,dt=\int _{\varphi ^{-1}(a)}^{\varphi ^{-1}(b)}f(\varphi (s))\cdot \varphi '(s)\,ds\,$

holds.

Proof, Write another proof