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
$$x=\phi (t)$$ 
a new variable $t$ satisfying

•
$\phi (\alpha )=a,\phi (\beta )=b$,

•
$\phi $ and ${\phi}^{\prime}$ are continuous on the interval with endpoints $\alpha $ and $\beta $.
Then
$${\int}_{a}^{b}f(x)\mathit{d}x={\int}_{\alpha}^{\beta}f(\phi (t)){\phi}^{\prime}(t)\mathit{d}t.$$ 
Proof. As a continuous function, $f$ has an antiderivative $F$. Then the composite function^{} $F\circ \phi $ is an antiderivative of $(f\circ \phi )\cdot {\phi}^{\prime}$, since by the chain rule^{} we have
$$\frac{d}{dt}F(\phi (t))={F}^{\prime}(\phi (t)){\phi}^{\prime}(t)=f(\phi (t)){\phi}^{\prime}(t).$$ 
Using the Newton–Leibniz formula (http://planetmath.org/node/40459) we obtain
$${\int}_{a}^{b}f(x)\mathit{d}x=F(b)F(a)=F(\phi (\beta ))F(\phi (\alpha ))={\int}_{\alpha}^{\beta}f(\phi (t)){\phi}^{\prime}(t)\mathit{d}t,$$ 
Q.E.D.
Title  change of variable in definite integral 
Canonical name  ChangeOfVariableInDefiniteIntegral 
Date of creation  20140527 13:13:22 
Last modified on  20140527 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 