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x-x-dx-




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

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