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\title{The Sophomore's Dream}
\author{Manuel Eberl}
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\begin{abstract}
This article provides a brief formalisation of the two equations known as the \emph{Sophomore's Dream}, first discovered by Johann Bernoulli~\cite{bernoulli} in 1697:
\[\int_0^1 x^{-x}\,\text{d}x = \sum_{n=1}^\infty n^{-n} \quad\text{and}\quad \int_0^1 x^x\,\text{d}x = -\sum_{n=1}^\infty (-n)^{-n}\]
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