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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.


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