That is, say someone posts a question about arithmetic geometry that I don't understand or can't answer, but I see that you (Pete L. Clark) have voted to close it, I will probably vote to close it as well.

Certainly I agree that there are many areas of mathematics in which it is easy to generate long (even infinite) lists of questions each of which will take a lot of work to answer. Currently we're talking about closed form antiderivatives, but Ryan mentioned giving finite presentations of groups and asking if they present the trivial group. I am more familiar with the latter problem but still not an expert, so in trying to figure out what I thought I quickly switched to yet a third class of problems: Diophantine equations. Note that a common feature of the latter two problems is that they are provably undecidable in general, but it is not (yet?) clear exactly or even approximately where the boundary is between un/decidability: if I give you a presentation with a "small" number of generators and relations, or say a single Diophantine equation of "small" degree, we think there should probably be answers. But it is going to be a lot of work!

Anyway, it was easy for me to figure out what I would do if someone posted several problems asking for solutions to various systems of Diophantine equations without any motivation. (It helps that this is essentially what I actually do when such questions come my way, which they do in real life.) Namely, I look at the system of equations quickly to see if (i) I recognize it as being of a very particular form that is well studied (e.g. a K3 surface, or a curve of genus one) or (ii) if it happens to catch my fancy for any reason. If neither (i) nor (ii) hold, then -- guess what? -- I don't answer the question or even think any more about it. On the other hand I don't complain about it either: I think "I have some random Diophantine equation, what can you make of it?" is a perfectly reasonable question, just not necessarily a very interesting question to me or, necessarily, any other arithmetic geometer.

By now you can probably see my point: just because you don't want to answer a question or are not interested in it does not mean it should be closed. As people have said, asking a question is an invitation for someone else to put in time and effort on your behalf, and no one is obligated to take up that invitation. But even if an invitation is not particularly, um, inviting, so long as it is clear and not too easy I think it is at least fair to ask it on MO.

Having looked back at some of these questions, um, in question, it seems that more clarity on exactly what kind of solution is sought could be a factor in their appropriateness on MO. Not being anything like an expert on this area, I had been assuming that what the rules were for closed form solutions were more or less agreed upon by experts in the area. But, although I don't think you need to motivate your question if you don't want to, if the community finds it to be unclear then of course the burden is upon you to clarify.

Finally, a lot of times people talk about "flooding the site" with questions of a particular type. Frankly I think this has never happened on MO. (The closest is when someone goes on a late night retagging binge, but once you realize that's what's happening it's not really problematic.) The SE platform is designed to negotiate a much higher level of traffic than our site has ever actually received: c.f. StackOverflow. To the best of my knowledge there have not been any more than, say, ten questions on indefinite summation. Am I wrong about this? If not, this is not flooding. If someone posted, say, five or more "appropriate, but not inviting" questions a day and continued this for several days heedless of a complete lack of response, then we would have something to talk about. But again, this is an yet purely hypothetical situation -- isn't it?

The important thing to remember is that by asking a question on MO, you are asking someone to do your work for you. The hope and intention is that the question would have taken you a day or more to solve but to the right person, just takes five minutes. But even that five minutes is something. People only have a finite number of "five minutes" to spend on MO (or at least, they should!) and they have to decide how to spend it. So putting in motivation is like the shop window: it says, "Hey, look at my question!"; it says, "Look how grateful I'll be if you answer it!"; it says, "Look how useful that answer will be to me, and all the wonderful things I'll be able to do with it!".

Motivation should answer a very simple question:

Why are you asking this question, now?

With the emphasis on "you" and the "now". What led you personally to this question, and why is now the time that you've turned to MO to help you find an answer.

I am not sure I share Pete's feeling that "perfectly random, purely technical question in some branch of mathematics" are to be encouraged, in the sense that although I've asked at least two questions which turned out in hindsight to be daft, I tried to give some indication that I had thought about the questions and why I thought that people might be able to give answers. I find myself inclined to side with Ryan on this one.

In general, I think that questions do not have to be motivated per se to be appropriate for MO. (I believe this sort of thing was discussed on meta before and this position was "agreed on", in the sense that everyone agreed except those who wrote in to disagree.) They just need to be of interest to some research mathematicians. Of course giving motivation is a good way of exhibiting and creating such interest (and perhaps one wants questions to be "good" and not just "appropriate"; for that motivation seems much more key). But I feel that I (for instance) should be able to ask a perfectly random, purely technical question in some branch of mathematics as long as it's not too easy, i.e., so that it takes a fairly expert person to answer it.

So if people are asking unmotivated indefinite integration questions that everybody knows how to routinely answer (in particular this should probably mean that the state of the art software packages like Mathematica can correctly answer it), then that's probably not appropriate for MO. Otherwise, I would say that it probably is.