change of variable in definite integral

Theorem.  Let the real functionxf(x)  be continuousMathworldPlanetmath on the interval[a,b].  We introduce via the the equation


a new variable t satisfying

  • φ(α)=a,φ(β)=b,

  • φ and φ are continuous on the interval with endpoints α and β.



Proof.  As a continuous function, f has an antiderivative F.  Then the composite functionMathworldPlanetmath Fφ is an antiderivative of (fφ)φ, since by the chain ruleMathworldPlanetmath we have


Using the Newton–Leibniz formula ( we obtain



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