Often when asking about a regularized value of an integral or series, I encounter a negative reaction of the sorts that "regularization is what you define it".


But in practice if we consider some widely used regularization methods: analytic continuation, Cesaro, Abel, Borel, Ramanujan etc we always get the same results where more than one method is applicable.


So why we cannot talk about THE regularization rather than consider it not-well-defined thing?


For instance, consider the following integral:


$$\operatorname{reg} \int\_{0}^{1}\frac{e^{1/x}}{x^2}\,dx=-e$$


I bet this regularized value is the natural regularization of this integral, regardless what (mutually consistent) methods one choses to apply (in this case one can come to the value via analytic continuation of $\int\_0^1 \frac{e^{a/x}}{x^2} \, dx$ over $a$).



Why we cannot speak about the main or natural regularization?
**Preliminary comment:** I second the idea that, for certain integrals, there should be a "well-behaved" class of regularization methods which all give the same value - of course, the important thing then is to give a good description of such a class of regularizations. 


**Answer to your question:** Yet, your claim that we can always agree on THE natural regularization is certainly too bold. I can't resist the sweet temptation to beat you with your own example ;-):


* The regularization which you suggest for the integral $\int\_0^1 \frac{e^{1/x}}{x^2} \,dx$ yields the value $-e$.
* Now consider the integral $\int\_0^1 e^{x^a}x^{a -1} \,dx$. For, say, $a > 0$ this yields $\frac{e-1}{a}$, which has a (unique) analytic continuation with respect to $a$ in $\mathbb{C} \setminus \{0\}$. If we choose $a = -1$, we obtain the value $1-e$, though.


