# change of variable in definite integral

Theorem.  Let the real function$x\mapsto f(x)$  be continuous on the interval$[a,\,b]$.  We introduce via the the equation

 $x\;=\;\varphi(t)$

a new variable $t$ satisfying

• $\varphi(\alpha)\,=\,a,\quad\varphi(\beta)\,=\,b$,

• $\varphi$ and $\varphi^{\prime}$ are continuous on the interval with endpoints $\alpha$ and $\beta$.

Then

 $\int_{a}^{b}\!f(x)\,dx\;=\;\int_{\alpha}^{\beta}\!f(\varphi(t))\,\varphi^{% \prime}(t)\,dt.$

Proof.  As a continuous function, $f$ has an antiderivative $F$.  Then the composite function $F\circ\varphi$ is an antiderivative of $(f\circ\varphi)\cdot\varphi^{\prime}$, since by the chain rule we have

 $\frac{d}{dt}F(\varphi(t))\;=\;F^{\prime}(\varphi(t))\,\varphi^{\prime}(t)\;=\;% f(\varphi(t))\,\varphi^{\prime}(t).$

Using the Newton–Leibniz formula (http://planetmath.org/node/40459) we obtain

 $\int_{a}^{b}\!f(x)\,dx\;=\;F(b)-F(a)\;=\;F(\varphi(\beta))-F(\varphi(\alpha))% \;=\;\int_{\alpha}^{\beta}\!f(\varphi(t))\,\varphi^{\prime}(t)\,dt,$

Q.E.D.

 Title change of variable in definite integral Canonical name ChangeOfVariableInDefiniteIntegral Date of creation 2014-05-27 13:13:22 Last modified on 2014-05-27 13:13:22 Owner pahio (2872) Last modified by pahio (2872) Numerical id 10 Author pahio (2872) Entry type Theorem Classification msc 26A06 Synonym change of variable in Riemann integral Related topic RiemannIntegral Related topic SubstitutionForIntegration Related topic FundamentalTheoremOfCalculus Related topic IntegralsOfEvenAndOddFunctions Related topic OrthogonalityOfChebyshevPolynomials