Definite Integral from 0 to 1 of x to the x
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Theorem
\(\ds \int_0^1 x^x \rd x\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n^n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\sum_{n \mathop = 1}^\infty \paren {-n}^{-n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0.78343 \ 05107 \ 12 \ldots\) |
Proof
We can write:
\(\ds x^x\) | \(=\) | \(\ds \map \exp {x \ln x}\) | Definition of Power to Real Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {x^n \paren {\ln x}^n} {n!}\) | Definition of Exponential Function |
So:
\(\ds \int_0^1 x^x \rd x\) | \(=\) | \(\ds \int_0^1 \paren {\sum_{n \mathop = 0}^\infty \frac {x^n \paren {\ln x}^n} {n!} }\rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\int_0^1 x^n \paren {\ln x}^n \rd x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\frac {\paren {-1}^n \map \Gamma {n + 1} } {\paren {n + 1}^{n + 1} } }\) | Definite Integral from $0$ to $1$ of $x^m \paren {\ln x}^n$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {n + 1}^{n + 1} }\) | Gamma Function Extends Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n^n}\) | shifting the index and using $\paren {-1}^{n + 1} = \paren {-1}^2 \paren {-1}^{n - 1} = \paren {-1}^{n - 1}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds -\sum_{n \mathop = 1}^\infty \paren {-\frac 1 n}^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\sum_{n \mathop = 1}^\infty \paren {-n}^{-n}\) |
Numerical computation of partial sums gives the decimal approximation.
$\blacksquare$