Two curious “identities” on xxx^x, ee, and π\pi

A numerical calculation on Mathematica shows that

I1=10xx(1x)1xsinπxdx0.355822

and

I2=10xx(1x)x1sinπxdx1.15573

A furthur investigation on OEIS (A019632 and A061382) suggests that I1=πe24 and I2=πe (i.e., |I1πe24|<10100 and |I2πe|<10100).

I think it is very possible that I1=πe24 and I2=πe, but I cannot figure them out. Is there any possible way to prove these identities?

Answer

You made a very nice observation! Often it is important to make a good guess than just to solve a prescribed problem. So it is surprising that you made a correct guess, especially considering the complexity of the formula.

I found a solution to the second integral in here, and you can also find a solution to the first integral at the link of this site.

Attribution
Source : Link , Question Author : zy_ , Answer Author : Sangchul Lee

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