$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic}{\mathrm{i}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mrm}[1]{\mathrm{#1}}
\newcommand{\pars}[1]{\left(\,{#1}\,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
\newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
\newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
\newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
$\ds{\bbox[15px,#ffd]{\int_{0}^{\infty}\expo{-x}\verts{\sin\pars{x}}\dd x = {1 \over 2}\,{\expo{\pi} + 1 \over \expo{\pi} - 1} =
{1 \over 2}\coth\pars{\pi \over 2}}:\ {\large ?}}$
\begin{align}
&\bbox[15px,#ffd]{\int_{0}^{\infty}\expo{-x}\verts{\sin\pars{x}}\dd x} =
\int_{0}^{\infty}\expo{-x}\mrm{sgn}\pars{\sin\pars{x}}\cos\pars{x}\,\dd x
\\[5mm] = &\
\int_{x\ =\ 0}^{x\ \to\ \infty}\mrm{sgn}\pars{\sin\pars{x}}
\,\dd\braces{{1 \over 2}\expo{-x}\bracks{\sin\pars{x} - \cos\pars{x}}}
\\[5mm] = &\
-\,{1 \over 2} - \int_{0}^{\infty}\braces{{1 \over 2}\expo{-x}
\bracks{\sin\pars{x} - \cos\pars{x}}}
\bracks{2\delta\pars{\sin\pars{x}}\cos\pars{x}}\,\dd x
\\[5mm] = &\
-\,{1 \over 2} + \int_{0}^{\infty}\expo{-x}\cos^{2}\pars{x}
\,\delta\pars{\sin\pars{x}}\,\dd x
\\[5mm] = &\
\sum_{n = -\infty}^{\infty}\int_{0}^{\infty}\expo{-x}\cos^{2}\pars{x}
\,\delta\pars{x - n\pi}\,\dd x
\\[5mm] = &\
-\,{1 \over 2} + \sum_{n = 0}^{\infty}\expo{-n\pi} =
-\,{1 \over 2} + {1 \over 1 - \expo{-\pi}} =
\bbox[15px,#ffd,border:1px solid navy]{{1 \over 2}\coth\pars{\pi \over 2}}\
\approx\ 0.5452
\end{align}