Integral from 0 to 1 (ln(1/x))^b dx

\begin{align*}& \int_0^1 \ln^b{(x)} dx = (-1)^b (b)! \\& \\&{\rm{Proof}}: \\& \\& \therefore x=e^u \Longrightarrow \int_{-\infty}^0 u^b e^u du \\& \\&\therefore u=-g \Longrightarrow (-1)^b \int_0^{\infty} g^b e^{-g} dg \\& \\& \therefore (-1)^b \int_0^{\infty} g^b e^{-g} dg = (-1)^b (b)! \\& \\&\therefore \int_0^1 \left( -\ln{(x)} \right)^b dx = (b)! \\& \\& \therefore \int_0^1 \ln^b{ \left( \dfrac{1}{x} \right) } dx = (b)! \\& \\& {\rm{Random \: Fact}}: \\& \\& \pi=\lim_{k \rightarrow \infty} \left[ \dfrac{2}{k} \sum_{n=1}^{k-1} \left( \ln{ \left(\dfrac{k}{n} \right)} \right)^{\frac{1}{2}} \right]^2 \\& \\&k=10^4 \Longrightarrow \pi \approx 3.14041 \Longrightarrow E_{\pi} \approx 0.0012
\end{align*}

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