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This question already has an answer here:

Question: $\ $Prove or disprove that $$\int_0^1\sin(\pi x)\,x^x\,(1-x)^{1-x}\ dx\ =\ \frac{e\pi}{4!}$$

The numerical integration by Matlab showed that $$\left|\ \int_0^1\sin(\pi x)\,x^x\,(1-x)^{1-x}\ dx\ - \ \frac{e\pi}{4!}\ \right|\ <\ 10^{-15}$$ which made me kind of believe that the equality holds.

Therefore, I tried to prove it by contour integration, integration by parts and all other methods I could come up with. However, it seems that none of them would work.

Could anyone help me with this question? Any hint will be much appreciated.

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marked as duplicate by Zaid Alyafeai, Community Apr 1 '17 at 20:42

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ a numerical method shows that this could be true $\endgroup$ – Dr. Sonnhard Graubner Apr 1 '17 at 14:40
  • $\begingroup$ @Dr.SonnhardGraubner Thanks! But do you think it is possible to prove it non-numerically? $\endgroup$ – Mengchun Zhang Apr 1 '17 at 14:44
  • $\begingroup$ I think it is possible at this time I have no idea $\endgroup$ – Dr. Sonnhard Graubner Apr 1 '17 at 14:46
  • $\begingroup$ According to WA, this is correct up to the 157th digit. No idea how to prove it either $\endgroup$ – Bananach Apr 1 '17 at 18:36
  • $\begingroup$ You've got much higher precision than I did, so now this is very likely to be true :) $\endgroup$ – Mengchun Zhang Apr 1 '17 at 19:03