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For a positive integer $n$ the terms given by \begin{align} & -\int_0^1 x^n \sin(\pi x) x^x (1-x)^{1-x} \, dx \\[8pt] = {} & \int_0^1\int_0^1 (xy)^n \sin(\pi xy) (xy)^{xy} \frac{(1-xy)^{1-xy}}{\ln (xy)} \, dx \, dy \end{align} are $r_n e\pi$ where $r_n$ are rationals. I wanted to know if it's possible to conjecture the terms given by $$\int_0^1\int_0^1 x^py^q \sin(\pi xy) (xy)^{xy}(1-xy)^{1-xy} \, dx \, dy $$ where $p,q$ are positive integers. In the first integral we can change $x^n$ by a Legendre Polynomial and try to prove that $e\pi$ is irrational, but only one polynomial is not enough to reach a contradiction. In the double integral above we can use two polynomials, but it's only useful if the integral converges to something familiar when $p\neq q$.

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  • $\begingroup$ Please don't use display mode in the title. I have edited to remove it. $\endgroup$
    – LSpice
    Commented Feb 21 at 22:59
  • $\begingroup$ Related mathoverflow.net/questions/226875/… $\endgroup$
    – Nemo
    Commented Feb 22 at 6:09

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