Definite Integral from 0 to 1 of x to the minus x

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

This sequence is A073009 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


We can write:

\(\ds x^{-x}\) \(=\) \(\ds \map \exp {-x \ln x}\) Definition of Power to Real Number
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-x}^n \paren {\ln x}^n} {n!}\) Definition of Exponential Function


\(\ds \int_0^1 x^{-x} \rd x\) \(=\) \(\ds \int_0^1 \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^n \paren {\ln x}^n} {n!} }\rd x\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {n!} \paren {\int_0^1 x^n \paren {\ln x}^n \rd x}\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {n!} \paren {\frac {\paren {-1}^n \map \Gamma {n + 1} } {\paren {n + 1}^{n + 1} } }\) Definite Integral from $0$ to $1$ of $x^m \paren {\ln x}^n$
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac 1 {\paren {n + 1}^{n + 1} }\) Gamma Function Extends Factorial
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty n^{-n}\) shifting the index

Numerical computation of partial sums gives the decimal approximation.