TITLE: Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$
QUESTION [8 upvotes]: $$\int_0^1 \frac{\arcsin(x)}{x}dx$$
This is a proposed for a Calculus II exam, and I have absolutely no idea how to solve it. Tried using Frullani or Lobachevsky integrals, or beta and gamma functions, but I can't even find a way to start it. Wolfram Alpha gives a kilometric solution, but I know that cannot be the only answer. Any help appreciated!

REPLY [5 votes]: $\newcommand{\+}{^{\dagger}}
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$\ds{\int_{0}^{1}{\arcsin\pars{x} \over x}\,\dd x:\ {\large ?}}$

\begin{align}&\color{#66f}{\large\int_{0}^{1}{\arcsin\pars{x} \over x}\,\dd x}
=-\int_{0}^{1}{\ln\pars{x} \over \root{1 - x^{2}}}\,\dd x
=-{1 \over 4}\int_{0}^{1}x^{-1/2}\ln\pars{x}\pars{1 - x}^{-1/2}\,\dd x
\\[3mm]&=-{1 \over 4}\lim_{\mu \to -1/2}\partiald{}{\mu}
\int_{0}^{1}x^{\mu}\pars{1 - x}^{-1/2}\,\dd x
=-{1 \over 4}\lim_{\mu \to -1/2}\partiald{}{\mu}
\bracks{\Gamma\pars{\mu + 1}\Gamma\pars{1/2} \over \Gamma\pars{\mu + 3/2}}
\\[3mm]&=-{1 \over 4}\,\Gamma\pars{\half}\braces{%
{\Gamma\pars{1/2} \over \Gamma\pars{1}}\,\bracks{%
\overbrace{\Psi\pars{\half}}^{\ds{-\gamma - 2\ln\pars{2}}}\ -\
\overbrace{\Psi\pars{1}}^{\ds{-\gamma}}}}\ =\
\half\,\Gamma^{2}\pars{\half}\ln\pars{2}
\\[3mm]&=\color{#66f}{\large\half\,\pi\ln\pars{2}}
\quad\mbox{with}\quad\Gamma\pars{\half} = \root{\pi}\quad\mbox{and}\quad\Gamma\pars{1} = 1.
\end{align}