I have searched the internet for an exact answer and although I have found many decimal approximations, https://www.wolframalpha.com/input/?i=integrate+x%5E(x)+from+0+to+infinity I have not been able to find an exact form.
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closed as offtopic by Andrés E. Caicedo, Alexandre Eremenko, Emil Jeřábek, Steven Landsburg, Nik Weaver Jun 12 at 14:16
This question appears to be offtopic. The users who voted to close gave this specific reason:
 "This question does not appear to be about research level mathematics within the scope defined in the help center." – Andrés E. Caicedo, Alexandre Eremenko, Steven Landsburg, Nik Weaver

1$\begingroup$ I doubt that there's a closed form. For $\int_0^1 x^{x} \, dx$ there's the memorable formual $\sum_{n=1}^\infty n^{n}$ [proof sketch: write $x^{x} = \exp(x\log x)$, use $e^t = \sum_{m=0}^\infty t^m/m!$, and integrate termwise]; but that's not really a closed form either. $\endgroup$ – Noam D. Elkies Jun 12 at 14:13