$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{{\displaystyle #1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic}{\mathrm{i}}
\newcommand{\on}[1]{\operatorname{#1}}
\newcommand{\pars}[1]{\left(\,{#1}\,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
\newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
\newcommand{\sr}[2]{\,\,\,\stackrel{{#1}}{{#2}}\,\,\,}
\newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
\newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
\begin{align}
& \color{#44f}{\int_{-\infty}^{\infty}{n \choose x}\dd x} =
\int_{-\infty}^{\infty}\bracks{\int_{-\pi}^{\pi}{\pars{1 + \expo{\ic\phi}}^{n} \over \expo{\ic x\phi}}{\dd\phi \over 2\pi}}\dd x
\\[5mm] = & \
\int_{-\pi}^{\pi}{\pars{1 + \expo{\ic\phi}}^{n}\,\
\overbrace{\int_{-\infty}^{\infty}\expo{-\ic\phi x}\,
{\dd x \over 2\pi}}^{\ds{\delta\pars{\phi}}}}\,\
\,\dd\phi = \pars{1 + \expo{\ic 0}}^{n} = \bbx{\color{#44f}{2^{n}}} \\ &
\end{align}