How do I understand if an indefinite integral is not solvable in terms of elementary functions?

Asked Aug 12 '20 at 12:05

Active Aug 12 '20 at 12:07

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Many indefinite integrals cant be solved in terms of elementary functions.e.g.-$$\int \frac{\sin(x)}{x}dx$$ But many hard looking ones are still solvable by weird substitutions and other tricks. Is there a test(like there are tests of solvability/convergence.etc in mathematics) of whether it is solvable or not by which I can see beforehand without wasting hours of time after which I come to know that it is not even solvable?

edited Aug 12 '20 at 12:07

José Carlos Santos

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asked Aug 12 '20 at 12:05

Manas Dogra

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This kind of question falls into the domain of "Differential Galois theory." The answer is given by Liouville's theorem. A more thorough presentation is given here. – Ben Grossmann Aug 12 '20 at 12:11
https://math.stackexchange.com/questions/155/how-can-you-prove-that-a-function-has-no-closed-form-integral
Duplicate
– Manas Dogra Aug 12 '20 at 12:22
@BenGrossmann: unfortunately, this material is inaccessible to most mortals :-( – Aug 12 '20 at 12:36

How do I understand if an indefinite integral is not solvable in terms of elementary functions?

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