"Values" of divergent integrals 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.
 A: 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
A: As I recall from QFT class the basic idea is to set the bound of integration to be $L$ and let $L \to \infty$.  
$$ \int_{-L}^L dx = 2L \quad\text{ and }\quad \int_{-L}^L x \, dx = L^2 $$
The integrals in physics are badly divergent.  The usual way is to set a parameter $\epsilon$   And there is a lot of talk about this method or that method being the canonical way of assigning a value.  
Dual to these are badly oscillatory integrals.  Revolving around the idea that $\sin n x \to 0$ is weakly convergent in $L^2[-\pi, \pi]$.  And  physicist might extract this limit exists generally over $\mathbb{R}$.

I think there is too much attention drawn to zeta function regularization  which is intimately  involved with the eq $\zeta(-1) = - \frac{1}{12}$.  As you note there are careful, it's possible to illustrate two plausible values for the same diveergent integrals and then you are in trouble.


*

*Henle & Kleinberg Infinitesimal Calculus

*John Bell A Primer of Infinitesimal 
And the warning here is they use two different types of infinesimals.  As I learned... one uses non-standard analysis, the other uses smooth infinitesimal analysis.  In particular, there's difficulty with the mean value theorem.

I have a feeling you know much more than this but I am not sure what to recommend.  Lately there is the fascinating theory of resurgence.  As one might know:
$$ \log n! = \sum_{k=1}^n \log k \approx \int_1^n \log x \, dx = n \log n - n $$
However if you try to "correct" this series too much, the answer on the right hand side is horribly divegent.  And this is tied to essential singularities such as $e^{-1/z}$.
A: Using [this approach][1] the values of these integrals are as follows:
$$\int_0^\infty 1\, dx = \omega_-+1/2=\tau$$
Its standard part (analog of regularization in case of series) is zero.
Here $$\omega_-=\sum_{k=1}^\infty 1$$,
the quantity of natural numbers, while $\tau$ is the quantity of all even or all odd numbers, half of the quantity of all integers.
$$\int_0^\infty \sin x\,dx=1$$
$$\int_0^\infty \cos x\,dx=0$$
The standard part of arbitrary improper integral of an analytic function can be calculated from this rule from divergent series:
$$\operatorname{st}\int_0^\infty f(x)\,dx=\lim_{s\to0} \operatorname{st} s\sum_{k=1}^\infty f(sk)$$
The following Mathematica code does the trick:
Sum[f[s x],{x,1,Infinity},Regularization->"Borel"]//FullSimplify
Limit[s %,s→0]

Thus,
$$\operatorname{st} \int_0^\infty e^x\, dx=-1$$
which coincides with analytic continuation, by the way.
