Definite Integral from 0 to 1 of x to the minus x
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Theorem
\(\ds \int_0^1 x^{-x} \rd x\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty n^{-n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1.29128 \ 5997 \ldots\) |
This sequence is A073009 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
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 |
So:
\(\ds \int_0^1 x^{-x} \rd x\) | \(=\) | \(\ds \int_0^1 \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^n \paren {\ln x}^n} {n!} }\rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {n!} \paren {\int_0^1 x^n \paren {\ln x}^n \rd x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {n!} \paren {\frac {\paren {-1}^n \map \Gamma {n + 1} } {\paren {n + 1}^{n + 1} } }\) | 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.
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Miscellaneous Definite Integrals: $15.119$