Are there existing theories of integration in which $I\_0 = \int\_0^{\infty} dx$ and $I\_1 = \int\_0^{\infty} x \ dx$ are well-defined infinite elements in a non-archimedean extension of the reals? I can think of at least two different ways to set up such a theory; in one, $I\_1 = I\_0^2$, and in the other, $I\_1 = I\_0^2/2$. (Both are based on mollification. In the first framework, you introduce a mollified version of $dx$, namely $e^{-\lambda x} dx$, and you look at the behavior of the mollified integral for $\lambda$ near $0^+$; in the second case, you think of the integral as the area under the graph, and you mollify in the $y$-direction as well.) There are no problems as long as the integrand is sufficiently nice, e.g., a polynomial function. I'm wondering if any such theories have been written down and/or been found to be useful.



"Values" of divergent integrals
For an attempt of such a theory, see <http://carlossicoli.free.fr/B/Burgin_M.-Hypernumbers_and_Extrafunctions__Extending_the_Classical_Calculus-Springer(2012).pdf>
(Hypernumbers and Extrafunctions: Extending the Classical Calculus, by Mark Burgin).


Link in amazon:
[http://www.amazon.com/Hypernumbers-Extrafunctions-Extending-SpringerBriefs-Mathematics/dp/1441998748](http://rads.stackoverflow.com/amzn/click/1441998748)


