integration under integral sign
Let
where is continuous in the rectangle
Then is continuous and hence integrable (http://planetmath.org/RiemannIntegrable) on the interval ; we have
This is a double integral over a in the -plane, whence one can change the order of integration (http://planetmath.org/FubinisTheorem) 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
we may interprete the integrand as a definite integral:
Accordingly, we can calculate as follows: