Sophomore's Dream

In mathematics, the Sophomore's dream is the pair of identities (especially the first) 

$$\begin{array}{rlrl}& {\int }_0^1{x}^{-x}dx& & =\sum _{n=1}^{\mathrm{\infty }}{n}^{-n}\\ & {\int }_0^1{x}^{x}dx& & =\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n+1}{n}^{-n}=-\sum _{n=1}^{\mathrm{\infty }}(-n{)}^{-n}\end{array}$$

discovered in 1697 by Johann Bernoulli.

The numerical values of these constants are approximately 

$1.291285997...$ and $0.7834305107...,$ 

respectively.

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