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\begin{align}&\color{#66f}{\large%
\int_{0}^{1}\left\lfloor{1 \over x}\right\rfloor^{-1}\,\dd x}
=\int_{\infty}^{1}\left\lfloor x\right\rfloor^{-1}\,\pars{-\,{\dd x \over x^{2}}}
=\int_{1}^{\infty}{\dd x \over \floor{x}x^{2}}
\\[5mm]&=\lim_{N \to \infty}\bracks{%
\int_{1}^{2}{\dd x \over x^{2}} + \int_{2}^{3}{\dd x \over 2x^{2}}+\cdots
+\int_{N - 1}^{N}{\dd x \over \pars{N - 1}x^{2}}}
\\[5mm]&=\lim_{N \to \infty}\bracks{%
\int_{0}^{1}{\dd x \over \pars{x + 1}^{2}}
+ \int_{0}^{1}{\dd x \over 2\pars{x + 2}^{2}} + \cdots
+\int_{0}^{1}{\dd x \over \pars{N - 1}\pars{x + N - 1}^{2}}}
\\[5mm]&=\lim_{N \to \infty}\int_{0}^{1}
\sum_{k = 1}^{N - 1}{1 \over k\pars{x + k}^{2}}\,\dd x
=-\lim_{N \to \infty}\int_{0}^{1}
\partiald{}{x}\sum_{k = 0}^{N - 2}{1 \over \pars{k + x + 1}\pars{k + 1}}\,\dd x
\\[5mm]&=-\int_{0}^{1}\partiald{}{x}\bracks{\Psi\pars{x + 1} - \Psi\pars{1}\over x}
\,\dd x
=-\bracks{\Psi\pars{2} - \Psi\pars{1} - \Psi'\pars{1}}
\\[5mm]&=\Psi'\pars{1} - 1 = \color{#66f}{\Large{\pi^{2} \over 6} - 1}
\approx {\tt 0.6449}
\end{align}
$\ds{\Psi\pars{z}}$ is the Digamma Function and we used the properties
\begin{align}
&\sum_{k = 0}^{\infty}{1 \over \pars{k + \mu}\pars{k + \nu}}
= {\Psi\pars{\mu} - \Psi\pars{\nu} \over \mu - \nu}\,,\qquad
\begin{array}{|rcl}
\ \Psi\pars{z + 1} & = & \Psi\pars{z} + {1 \over z}
\\
\Psi'\pars{1} & = & {\pi^{2} \over 6}
\end{array}
\end{align}