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

$${\int }_a^{b}f(x)𝑑x={\int }_{\alpha }^{\beta }f(\phi (t)){\phi }^{\prime }(t)𝑑t.$$

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

$${\int }_a^{b}f(x)𝑑x=F(b)-F(a)=F(\phi (\beta ))-F(\phi (\alpha ))={\int }_{\alpha }^{\beta }f(\phi (t)){\phi }^{\prime }(t)𝑑t,$$

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