Evaluate the Integral: $$\int_{-\pi}^{\pi}\ {e^{e^{\sin x}}}\ dx $$
I'm curious as to whether or not this integral has been solved and if solved how to go on about it.
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Why would you expect there to be a symbolic solution? – YiFan Tey Sep 15 '18 at 03:29
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I plan on solving it, just want to know if anyone has already done it first. – Nimish Sep 15 '18 at 03:57
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There might be a way of using complex analysis. – Tom Himler Sep 15 '18 at 04:05
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1Consider adding your thoughts and attempts in your question. – paulplusx Sep 15 '18 at 04:56
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Not all integrals are "solvable". Your solution is approximately $31.785$ as pointed out in Ross' answer, but there might not be a symbolic, analytical solution. It can be unsolvable. – YiFan Tey Sep 15 '18 at 05:10
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You may expand the outer exponential as a Maclaurin series, then exploit $\int_{-\pi}^{\pi}\exp(m\sin x),dx = 2\pi I_0(m) $ where $I_0$ is the principal modified Bessel function of the first kind. – Jack D'Aurizio Sep 15 '18 at 18:31
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You can feed it to Alpha, getting a result that say the integral is aobut $31.785$ I would be amazed if there is an analytic solution and expect that this is a numerical answer. The fact that an exact form was not offered supports this. It is easy to write integrals that cannot be solved analytically.
Ross Millikan
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