$$
\begin{align}
\int_0^\pi e^{\cos(x)}\cos(\sin(x))\,\mathrm{d}x
&=\frac12\int_{-\pi}^\pi e^{\cos(x)}\cos(\sin(x))\,\mathrm{d}x\\
&=\frac12\mathrm{Re}\left(\frac1i\int_{-\pi}^\pi e^{e^{ix}}e^{-ix}\,\mathrm{d}e^{ix}\right)\\
&=\frac12\mathrm{Re}\left(\frac1i\int_\gamma\frac{e^{u}}{u}\,\mathrm{d}u\right)
&&\begin{array}{l}
\text{$\gamma$ follows the unit circle}\\
\text{ counterclockwise from}\\
\text{$e^{-i\pi}$ to $e^{i\pi}$}
\end{array}\\[9pt]
&=\pi
\end{align}
$$