Integration/Substitution/Fact

Substitution rule

Suppose that ${}I$ denotes a real interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a continuous function. Let

$g\colon [a,b]\longrightarrow I$

be a continuously differentiable function.

Then

${}\int _{a}^{b}f(g(t))g'(t)\,dt=\int _{g(a)}^{g(b)}f(s)\,ds\,$

holds.

Proof, Write another proof