@amWhy Well I suppose this is one of the borderline questions which we may disagree upon. I do happen to think that the real content of the question is in fact on-topic, and falls under the tags soft-question and recreational-mathematics.
If I recall correctly, I did check the same help centre page linked by Jryki before answering, and decided that the underlying question does: (1) inspire answers that explain "why" or "how" these integrals were concocted; (2) tend to have long answers; (3) have a constructive tone; (4) invite sharing experiences and not just opinions. In particular, the question asks for mathematical strategies to produce 'nice' integrals, so although "nice" is subjective the actual strategies would be concrete.
My final reason is that on reading the question I recalled the examples in my post, and saw that they had not yet been mentioned, but felt that they were quite nice and deserved to be more known. However, I will just let other people decide whether they feel it should be closed. I see that it has been closed and reopened and may be closed again soon.
@AlexFrancisco @XanderHenderson: No there is no such tool. Once upon a time someone was very unhappy with terrible PSQs of that sort being closed and deleted, and complained that there were a whole lot of them, essentially "See this big list that I found/generated! Why aren't you doing anything about them?" and so we thanked that one for providing such a helpful list. =)
I will also add a ping for @robjohn - since he posted an answer to the original version. (There is one more answer there - however, it seems to me more like a hint than an actual answer.)
Question to the native speakers of the room. I feel that this edit does not correct grammatical issues, but rather introduces a few (with the exception of statement/'s). Should I reject it?
@user21820 It's you're option whether or not to choose to answer a question or not. I was quite surprised there. However, my surprise doesn't prohibit you from doing so.
@user21820 Limited as it is. It needs to be run regularly, and it needs to have conditions that other users can also participate in such a search, as well: more disqualifiers, perhaps more qualifiers, as well.
@amWhy Yes one still needs to read every post to determine whether it is really a bad question. I haven't the energy to craft a better query either, and I think one needs an account on that site. If you craft any better queries let us know! =)
@MikePierce Worked quite effectively. I liked the naming of the otherwise anonymous posts, which too often, are attributed to their careless authors. ( Maybe we should develop a pool of click bait, like "milking an integral" to better address crap ... oops, to better address "issues" in CRUDE or on META. Yesterday's clickbait, unfortunately: milking an integral, as though an integral is a cow!
What is there to do in logic Err, What is there to do in pie making, as we erase all the examples of both what we don't mean in logic nor in pie-making, until we virtually redefine pie making and logic.
@PeterSheldrick @DanielFischer, Care to elaborate on your comment to the question immediately above, as an answer to the question. You can take around 30 minutes!