Question Number 9158 by geovane10math last updated on 21/Nov/16
$$\int{x}^{{x}} \:{dx}\:=\:? \\ $$
Answered by FilupSmith last updated on 22/Nov/16
$${x}^{{x}} ={e}^{{x}\mathrm{ln}\left({x}\right)} \\ $$$${e}^{{x}\mathrm{ln}\left({x}\right)} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} \mathrm{ln}^{{n}} \left({x}\right)}{{n}!} \\ $$$$\Rightarrow\int{x}^{{x}} {dx}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}!}\int\left({x}\mathrm{ln}\left({x}\right)\right)^{{n}} {dx} \\ $$