Abstract
This article provides a brief formalisation of the two equations known as the Sophomore's Dream, first discovered by Johann Bernoulli in 1697:
\[\int_0^1 x^{-x}\,\text{d}x = \sum_{n=1}^\infty n^{-n} \quad\text{and}\quad \int_0^1 x^x\,\text{d}x = -\sum_{n=1}^\infty (-n)^{-n}\]