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For all I know, there are integrals which are not possible to solve - an example I was told is$$\int{\frac{\sin (x)}{x}}\,dx.$$ How to identify whether it has a closed form antiderivative or not? Is there a method?

Ian Mateus
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benjamin_ee
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    I'm having a bit of trouble understanding your question. Do you mean to ask if there is a method for determining whether or not an indefinite integral has solutions in terms of elementary functions? – Alex Wertheim Jul 30 '13 at 17:27
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    Check out the Risch_algorithm. This can check whether a function is integrable over a field of standard functions. – N3buchadnezzar Jul 30 '13 at 17:28
  • @AWertheim: Yes. – benjamin_ee Jul 30 '13 at 17:42
  • I think this is a duplicate of a extensively debated question previously asked. – Ian Mateus Jul 30 '13 at 18:44
  • @IanMateus That question revolves around the phrasing "closed form", the pedantry surrounding which need not introduce itself here. – Emily Jul 30 '13 at 18:54
  • @Arkamis I think this "pedantry" is the main point of the question. Would you consider $\Gamma$ as closed form? $\zeta$, perhaps? In some contexts they are, in others aren't, this distinction should be made explicit. I think that question perfectly answers this one, until OP comes up with a more specific question. – Ian Mateus Jul 30 '13 at 19:01
  • I think that "closed form" is mentioned nowhere in the question, and hence the debate is not germane here. – Emily Jul 30 '13 at 19:04
  • @Arkamis Fair enough, I have put "closed form" in the question, no one else. This is what I could understand from his phrasing, and I've put it in my edit. This is my interpretation (which I believe to transmit his doubts). If this is not the case, OP is always free to point it out. – Ian Mateus Jul 30 '13 at 19:11
  • @IanMateus I see no reason why we should include "closed form" in the OP when it cannot be determined to be the intention any more than "expressible in elementary functions", the latter of which has an answer, which myself and others have already pointed out. In any case, there is no need to transform this question into a duplicate of one that has already generated much debate, almost all of which is entirely subjective, and hence will go on indefinitely. – Emily Jul 30 '13 at 19:14

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The only general algorithm is the Risch algorithm. This is, in general, infeasible to apply by hand. Therefore the only way to identify whether an anti-derivative has a symbolic representation expressible in terms of elementary functions is to become familiar with a huge repertoire of tricks, identities, and techniques.

Emily
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If you know a bunch of integrals that don't have symbolic representation in terms of elementrary functions, then for a given integral you can inspect and try to see if there is a symbolic substitution that transforms a related-looking "impossible" integral into yours. If so, then you know you can't find a symbolic representation in terms of elementary functions for your integral either. Otherwise I agree, it takes computations that are too complicated by hand in general to decide.

user2566092
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