# Sophomore's Dream

## 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 $x^{-x}$

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