Sophomore's Dream

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Theorem

Sophomore's Dream refers to two identities discovered in 1697 by Johann Bernoulli.

Definite Integral from $0$ to $1$ of $x^x$

\(\ds \int_0^1 x^x \rd x\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n^n}\)
\(\ds \) \(=\) \(\ds -\sum_{n \mathop = 1}^\infty \paren {-n}^{-n}\)
\(\ds \) \(=\) \(\ds 0.78343 \ 05107 \ 12\ldots\)


Definite Integral from $0$ to $1$ of $x^{-x}$

\(\ds \int_0^1 x^{-x} \rd x\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty n^{-n}\)
\(\ds \) \(=\) \(\ds 1.29128 \ 5997 \ldots\)


Also see