The integration is meaningless if
$latex displaystyle {int x^{x} dx}&fg=00aa00&s=3$
unless specifying the domain of defintion, eg. interval [0,1] :
$latex displaystyle {int_{0}^{1} x^{x} dx = frac {pi^{2}}{12} }&fg=aa0000&s=3$
Note: Gamma function proof by Feynman trick : differentiate under integral.
DEMO:
Mathtuition88 