Sophomore's dream

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

$${\displaystyle {\begin{alignedat}{2}&\int _{0}^{1}x^{-x}\,dx&&=\sum _{n=1}^{\infty }n^{-n}\\&\int _{0}^{1}x^{x}\,dx&&=\sum _{n=1}^{\infty }(-1)^{n+1}n^{-n}=-\sum _{n=1}^{\infty }(-n)^{-n}\end{alignedat}}}$$ discovered in 1697 by Johann Bernoulli.

The numerical values of these constants are approximately 1.291285997... and 0.7834305107..., respectively.

The name "sophomore's dream" is in contrast to the name "freshman's dream" which is given to the incorrect identity ${\textstyle (x+y)^{n}=x^{n}+y^{n}}$. The sophomore's dream has a similar too-good-to-be-true feel, but is true.