Substitution rule (change of variables) for Riemann integrals

Substitution rule: Let $f:[a,b]\to\mathbb{R}$ be continuous , and let $\varphi:[\alpha,\beta]\to[a,b]$ be continuously differentiable and monotone on $[\alpha,\beta]$. Then $\int_\alpha^\beta f(\varphi(t))\,\varphi'(t)\,dt=\int_{\varphi(\alpha)}^{\varphi(\beta)} f(u)\,du.$ (If $\varphi$ is decreasing, the right-hand side automatically changes sign because $\varphi(\alpha)>\varphi(\beta)$.)

This formula is the rigorous justification for substitution in calculus and is the one-dimensional prototype for higher-dimensional change of variables .

Proof sketch: Let $F$ be an antiderivative of $f$ (possible since $f$ is continuous). Then $(F\circ\varphi)'(t)=f(\varphi(t))\varphi'(t)$. Apply the fundamental theorem of calculus : $\int_\alpha^\beta f(\varphi(t))\varphi'(t)\,dt = (F\circ\varphi)(\beta)-(F\circ\varphi)(\alpha)=F(\varphi(\beta))-F(\varphi(\alpha)),$ and rewrite the last expression as $\int_{\varphi(\alpha)}^{\varphi(\beta)} f(u)\,du$.