Should we do anything with the tons of seemingly impossible integrals?

Question

I've seen huge amounts of seemingly impossible integrals. A lot have no nice answer while some have very nice answers.

I don't want to remove these answers as many are (IMO) quite beautiful. However I do want to somehow contain these a bit, and keep quality up (some integrals can't be computed in any nice way). Should we do anything about these? Other than some really pretty answers, I don't see people getting too much out of it for future reference.

Would it be wise to have some giant CW thread?

Examples:

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$

Integral $ \int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx$

Integrate $ \int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} dx=8\pi \ln 2$

Integral $ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx$

Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

Answer 1

Should we do anything about these?

Tag them as (definite-integral) so that people who want to find them and try to answer can do that.

Upvote the ones that look more interesting, and their answers.

Read if interested and answer if able.

No other action is needed.

Other than some really pretty answers, I don't see people getting too much out of it for future reference.

You underestimate the value and mathematical content.

There are some errors in published tables of integrals, so it is good to work the harder ones on the internet from time to time.

Of the examples in the question, 4/5 are period integrals. Those have significant structure and a (still developing!) body of technique that can be used to evaluate them. A library of examples where this is done in nontrivial cases has long term value even for researchers in the field. It is not necessary that somebody want to compute exactly the integral posted, for the MSE question to have value.

This is the rare area of mathematics where amateurs using pencil and paper can do things that are of interest at the front line. It is also keeping alive and documenting the older style of formula-heavy calculus, and human expertise in that, which is important if you don't want old knowledge to be lost over time.

Answer 2

Would it be wise to have some giant CW thread?

No. Having multiple integrals in one question would make it too broad. Also, it makes no sense for it to be CW, denying reputation points to the users who provide answers, sometimes very nontrivial.

Other than some really pretty answers, I don't see people getting too much out of it for future reference.

The computations leading to those answers have ideas, that may be learned by studying them, and used elsewhere. At least, sometimes.

And "sometimes" is all that we can realistically expect, remembering Sturgeon's Law. Most of those integrals may be items of no consequence, but this does not make them different from the majority of questions asked in any given tag (or from the majority of theorems published in any particular area of mathematics).

Answer 3

Caveat: My opinions only. I have not discussed this with other moderators.

IMHO it is questions like this recent beauty that cause more resentment towards questions about integrals. Several answers, certainly nothing that has not been covered zillion times on the site - yet not a single answerer or passer by put in the effort to find a reasonable duplicate. It looks like high-volume freshman questions (not forgetting high school) simply disregard all rules about duplicates.

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Enough of that. Back to the question at hand.

I don't think that collecting these integrals to a single CW question would work. That would be a bad question, and, unless I am mistaken, the techniques are relatively diverse.

It could serve as a list of useful links - collecting pointers to techniques for the benefit of whoever wants to learn the tricks of the trade. However, such a list needs maintenance, i.e. a curator. If Ron Gordon or some other active user volunteers to maintain the list, I would certainly support the idea. That may be too much of a commitment, and the interested users can find these questions without such help. So it is not clear it is worth the trouble.

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However, there may be some valid concerns and misgivings about these questions. As I have had such feelings myself, let me share them and my conclusions with you. These are impressions of a casual outsider observer.

1. Sometimes I could not shake the feeling that those integrals were posted as challenges rather than genuine questions about a problem somebody had encountered. May be straight out of Gradshtein-Ryzhik or some such tome? I guess that this worry originates from Anastasia's past suggestion of actually running an integration contest. The idea was that the contest organizer would hand out bounties as prizes. This was shot down in a meta discussion at the time. So the concern was whether the idea has survived in the form of a continuously running integration bee :-). But I agree with the sentiment of Normal Human's answer (if I got it right). I don't really see any harm coming from such questions. At least not if these highly upvoted opinions are observed. Let's not forget that similar opportunities are available in other areas of math as well.
2. These tough integrals seem to be mostly about honing a skill set rather than reaching for new conceptual heights. But, again, A) where's the harm to the site, and B) similar opportunities exist elsewhere as well. It is those repetitive integrations by parts, standard substitutions, and questions about partial fractions expansions that should (IMVHO) fall under there are only 40 calculus questions.
3. There could be a concern that such questions and their answers all come from a smallish group of users, and thus they might silently form a voting ring. Perish this thought! Right away! As a diamond-bearer I can see some stats. Sure, the active integrators have upvoted each other, but the scale is not any different from what you see within any other group of users sharing an interest in some tag.

TL; DR; My misgivings seem to originate from jealousy. Boo-hoo. This is not my cup of tea, so these people should not gain so much rep from working out these esoteric integrals either. If we let the green-eyed monster form site policies I think a downward spiral is inevitable.

So if you don't like these integrals, just upvote other stuff instead.

Answer 4

I see that @NormalHuman has made the points I would normally make already.

However, as for the suggestion to close made in the comments, that is fine. But keep in mind the odds are stacked against those who think a question should genuinely remain open. As I see it, it merely takes 5 people (and for tags related to integration it is usually the same 5 people) to close a question, while there are frequently many more who would rather see such questions open.

Accordingly, I have a suggestion: holders of gold badges in a tag should have the right to unilaterally reopen a question in that tag, while those who vote to close do not get a second vote. (Right now, they have the right to unilaterally close a question as a duplicate.) If a gold-badge holder voted to close in the first place, then the standard reopen process remains. I feel that this will restore a little balance in how questions in certain tags are treated here.

I for one am exhausted from defending my right to solve interesting problems that have been a credit to the site.