I know that there are integrals that cannot be written in this way such as $e^{-x^2}$ but there is a general rule that can be used to recognize these types of integrals?
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There is a way, but often not easy to do by hand. A key ingredient is Liouville's theorem – NDB Jul 25 '23 at 14:08
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No. The only way I know of is to check the integral on WA or search the many lists on Wikipedia. The rest comes with experience. – Marius S.L. Jul 25 '23 at 14:09
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@NDB This looks as if it resulted in a power series. But how do we know whether the power series belongs to a known function or not? – Marius S.L. Jul 25 '23 at 14:13
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3@MariusS.L. Nothing to do with power series. The answer "No" is mathematically incorrect, if the coefficients are not too bad. – NDB Jul 25 '23 at 14:14
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Theorem: Given $f, g \in C(X)$ with $f \neq 0$ and $g \neq const$. $f(x)e^{ g(x)}$ can be integrated in elementary functions $\iff \exists R ∈ C(X)$ such that $$ R'(X) + g'(X)R(X) = f(X)$$ has a solution in $C(X)$.
$C(X)$ is essentially a field of rational functions.
There is a whole theory behind this, so the proof cannot be given in a single post here. If you want to delve deeper, you need a strong abstract algebra background, differential algebra specifically. Look into Liouville's theorem in particular.
J. W. Tanner
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