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单词 ChangeOfVariableInDefiniteIntegral
释义

change of variable in definite integral


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

x=φ(t)

a new variable t satisfying

  • φ(α)=a,φ(β)=b,

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

Then

abf(x)𝑑x=αβf(φ(t))φ(t)𝑑t.

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

ddtF(φ(t))=F(φ(t))φ(t)=f(φ(t))φ(t).

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

abf(x)𝑑x=F(b)-F(a)=F(φ(β))-F(φ(α))=αβf(φ(t))φ(t)𝑑t,

Q.E.D.

Titlechange of variable in definite integral
Canonical nameChangeOfVariableInDefiniteIntegral
Date of creation2014-05-27 13:13:22
Last modified on2014-05-27 13:13:22
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id10
Authorpahio (2872)
Entry typeTheorem
Classificationmsc 26A06
Synonymchange of variable in Riemann integral
Related topicRiemannIntegral
Related topicSubstitutionForIntegration
Related topicFundamentalTheoremOfCalculus
Related topicIntegralsOfEvenAndOddFunctions
Related topicOrthogonalityOfChebyshevPolynomials
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