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A few days back, the following integral was posted $$\int_0^1 x^x(1-x)^{1-x}\sin(\pi x)\,dx=\frac{\pi e}{24}$$

The integral was answered using complex analysis tools but I am interested in other methods which do not use residue calculus.

I am unable to come up with anything for this. I noticed that the integrand is symmetric about $x=1/2$ but this doesn't help much.

Help is appreciated. Thanks!

Git Gud
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Pranav Arora
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  • $\sin(\pi x)$ is asking for residue calculus. It would be unlikely for an elementary solution to exist. – Ali Caglayan Oct 22 '14 at 09:58
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    Just in case anybody is looking for that question: http://math.stackexchange.com/questions/958624/prove-that-int-01-sin-pi-xxx1-x1-x-dx-frac-pi-e24 – Hans Lundmark Oct 22 '14 at 11:58
  • It's possibly duplicate of this, since nobody said that I'm looking for an answer using residue methods. Anybody could answer the integral there using real analysis. Anyway I do not accepted the answer there, since it is just a partial answer because the half of the derivation is missing, furthermore there was a bounty on the question and nobody has posted other solutions. – user153012 Oct 23 '14 at 16:18

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