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?
2026-04-29 17:18:15.1777483095
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How do we know if an antiderivative has a closed form?
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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.
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.