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The integral $\int_{0}^{\infty} \cos(x) dx$ is considered to be divergence, as well as $ \int_{1}^{\infty} x^{-1} dx$.
Now I'm studying curves in space, and there are many integrals from curvature, torsion and their combinations.

It would be important for me to separate these types of behavior. Personally, I would call $ \int_{1}^{\infty} x^{-1} dx$ converge to $+ \infty $, and say $\int_{0}^{\infty} x \cos(x) dx$ converge to $\infty $ by absolute value.
$ \int_{0}^{ \infty} \cos(x) dx$ could be called something else: no limit, undefined, etc.

One can give a strict definition of "convergence to $ \infty $", like:
for any $L >0$ there is $x^* > 0$ such that $|\int_{0}^{s} f(x) dx|>L$ for any $s > x^*$

Are there any such definitions in the analysis? Maybe not in the rigorous courses, but did no one emphasize that this is a completely different behavior?

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  • $\begingroup$ Do you mean $\displaystyle\int_a^\infty f(x)\mathrm{d}x=+\infty$ and $\displaystyle\left\lvert\int_a^\infty f(x)\mathrm{d}x\right\rvert=+\infty$? $\endgroup$ Commented Oct 24 at 16:45
  • $\begingroup$ @JCQ, These types of behavior can also be divided, but for me is much more important to separate both from the "diverging" $ \int_{a}^{ \infty} \cos(x) dx$ $\endgroup$ Commented Oct 24 at 16:48
  • $\begingroup$ Yes I mean are these what you are looking for? They are just simply called "infinite limits", and this concept is widely used. You can check wiki entries involving concept of limits like limit of a function. $\endgroup$ Commented Oct 24 at 17:00

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