integration under integral sign



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


Then  αI(α)  is continuous and hence integrable ( on the intervalα1αα2;  we have


This is a double integral over a in the xα-plane, whence one can change the order of integration ( and accordingly write


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:


Accordingly, we can calculate as follows:

I =0(abe-αx𝑑α)𝑑x
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