Denote by \(\lambda\) one-dimensional Lebesgue measure on the interval \((0,1)\).

(i) Show that for all \(k \in \mathbb{N}_{0}\) one has

\[\int_{(0,1)}(x \ln x)^{k} \lambda(d x)=(-1)^{k}\left(\frac{1}{k+1}ight)^{k+1} \Gamma(k+1) .\]

(ii) Use (i) to conclude that
\[
\int_{(0,1)} x^{-x} \lambda(d x)=\sum_{k=1}^{\infty} k^{-k}
\]
[ note that \(x^{-x}=e^{-x \ln x}\) and use the exponential series.]