calculus

\int_{0}^{1}x^xdx=\int_{0}^{1}e^{x.lnx}dx=\int_{0}^{1}\sum_{n=0}^{\infty}\frac{(x.lnx)^n}{n!}dx=\sum_{n=0}^{\infty}\frac{1}{n!}.\int_{0}^{1}x^n.(lnx)^ndx = $\\\\\\\\ $=\sum_{n=0}^{\infty}.\frac{1}{n!}.\left \{ \frac{1}{n+1}.x^{n+1}.ln^nx-\frac{n}{(n+1)^2}.x^{n+1}.ln^{n-1}x......+\frac{(-1)^n.n!}{(n+1)^{n+1}}.x^{n+1} \right \}\Big|_{0}^{1}\Big$\\\\\\\\ =\sum_{n=0}^{\infty}\frac{1}{n!}.\frac{(-1)^n.n!}{(n+1)^{n+1}}.x^{n+1}=\sum_{n=1}^{\infty}\frac{1}{n^n}

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