\begin{alignedat}{2}&\int _{0}^{1}x^{-x}\,dx&&=\sum _{n=1}^{\infty }n^{-n}\\&\int _{0}^{1}x^{x}\,dx&&=\sum _{n=1}^{\infty }(-1)^{n+1}n^{-n}=-\sum _{n=1}^{\infty }(-n)^{-n}\end{alignedat}
discovered in 1697 by Johann Bernoulli.
The numerical values of these constants are approximately
$1.291285997... $ and $0.7834305107...,$
respectively.
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