integration under integral sign


Let

I(α)=abf(x,α)𝑑x.

where  f(x,α) is continuousMathworldPlanetmath in the rectangle

axb,α1αα2.

Then  αI(α)  is continuous and hence integrable (http://planetmath.org/RiemannIntegrable) on the intervalα1αα2;  we have

α1α2I(α)𝑑α=α1α2(abf(x,α)𝑑x)𝑑α.

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

α1α2(abf(x,α)𝑑x)𝑑α=ab(α1α2f(x,α)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=0e-ax-e-bxxdx  (a>0,b>0),

we may interprete the integrand as a definite integral:

e-ax-e-bxx=/α=bae-αxx=abe-αx𝑑α.

Accordingly, we can calculate as follows:

I =0(abe-αx𝑑α)𝑑x
=ab(0e-αx𝑑x)𝑑α
=ab(/x=0-e-αxα)𝑑α
=ab1α𝑑α=/ablnα
=lnba
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