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integration under integral sign


Let

I(Ξ±)=∫baf(x,Ξ±)𝑑x.

where  f(x,Ξ±) is continuousMathworldPlanetmath in the rectangle

a≦

Then  α↦I⁒(Ξ±)  is continuous and hence integrable (http://planetmath.org/RiemannIntegrable) on the interval  α1≦α≦α2;  we have

∫α1Ξ±2I⁒(Ξ±)⁒𝑑α=∫α1Ξ±2(∫abf⁒(x,Ξ±)⁒𝑑x)⁒𝑑α.

This is a double integral over a in the x⁒α-plane, whence one can change the order of integration (http://planetmath.org/FubinisTheorem) and accordingly write

∫α1Ξ±2(∫abf⁒(x,Ξ±)⁒𝑑x)⁒𝑑α=∫ab(∫α1Ξ±2f⁒(x,Ξ±)⁒𝑑α)⁒𝑑x.

Thus, a definite integral depending on a parametre may be integrated with respect to this parametre by performing the integration under the integral sign.

Example.  For being able to evaluate the improper integral

I=∫0∞e-a⁒x-e-b⁒xxdx  (a>0,b>0),

we may interprete the integrand as a definite integral:

e-a⁒x-e-b⁒xx=/Ξ±=ba⁑e-α⁒xx=∫abe-α⁒x⁒𝑑α.

Accordingly, we can calculate as follows:

I  =∫0∞(∫abe-α⁒x⁒𝑑α)⁒𝑑x
 =∫ab(∫0∞e-α⁒x⁒𝑑x)⁒𝑑α
 =∫ab(/x=0∞-e-α⁒xΞ±)⁒𝑑α
 =∫ab1α⁒𝑑α=/ab⁑ln⁑α
 =ln⁑ba
Title integration under integral sign
Canonical name IntegrationUnderIntegralSign
Date of creation 2013-03-22 18:46:27
Last modified on 2013-03-22 18:46:27
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Theorem
Classification msc 26A42
Related topic FubinisTheorem
Related topic DifferentiationUnderIntegralSign
Related topic RelativeOfExponentialIntegral