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I wonder if an elementary antiderivative of the function $e^{\sin x} \sin x$ exist? If so, could anyone help me to derive this certain antiderivative step by step? If not, is a strict proof of the nonexistence available, maybe by using knowledge of differential algebra (with which I'm not familiar)? Thanks in advance!

zyy
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  • No. https://www.wolframalpha.com/input?i=integral+e%5E%28sin%28x%29%29sin%28x%29 – morrowmh Mar 24 '22 at 17:42
  • @MichaelMorrow Thanks. But I still expect a strict proof showing the inexistence of the antiderivative. – zyy Mar 24 '22 at 17:45
  • See here: https://wstein.org/edu/winter06/20b/Conrad.pdf – operatorerror Mar 24 '22 at 18:04
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    Liouville's theorem sets criteria for the existence of anti-derivative https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra) – Vasili Mar 24 '22 at 18:44
  • @Vasya Thanks. It seems to me that there is still quite a distance between a preliminary understanding of Liouville's theorem and a successful application of it to a specific problem like this. – zyy Mar 25 '22 at 07:42
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    @operatorerror Thank you. I'll take some time to digest it. – zyy Mar 25 '22 at 07:43
  • @zyy: You may be interested in reading this excellent answer about application of the theorem: https://math.stackexchange.com/questions/265780/how-to-determine-with-certainty-that-a-function-has-no-elementary-antiderivative/265884#265884 – Vasili Mar 25 '22 at 16:48

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Write $$e^{\sin (x)}\, \sin (x)=\sum_{n=0}^\infty \frac{\sin ^{n+1}(x)}{n!}$$ $$\int e^{\sin (x)}\, \sin (x)\,dx=\sum_{n=0}^\infty \frac 1{n!}\int \sin ^{n+1}(x)\,dx$$ Use the reduction formula $$I_n=\int \sin ^{n}(x)\,dx \quad \implies \quad I_n= \dfrac {n - 1} n I_{n - 2} - \dfrac {\sin^{n - 1}( x) \cos (x)} n $$ and, for definite integrals, you can generate nice identities such as $$\int_0^{\frac \pi 2} e^{\sin (x)}\, \sin (x)\,dx=\sum_{n=0}^\infty \frac 1{n!} \frac{\sqrt{\pi } \,\,\Gamma \left(\frac{n}{2}+1\right)}{2\, \Gamma \left(\frac{n+3}{2}\right)}=\frac{\pi}{2} (\pmb{L}_{-1}(1)+I_1(1))$$ where appear Struve and Bessel functions.

Claude Leibovici
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