Cosine sine integral

Prove that
$$\int_0^{2\pi} \cos(\sin{x})e^{\cos{x}} dx = 2\pi$$

$$\textit{proof}$$

$$\begin{align}
\int_0^{2\pi} \cos(\sin{x})e^{\cos{x}} dx &=Re\left( \int_0^{2\pi} e^{i\sin(x)}e^{\cos{x}} dx\right) \\
&= Re \left(\int_0^{2\pi} e^{e^{i x}} dx\right)\\
&= Re \left(\sum_{n=0}^\infty \frac{1}{n!}\int_0^{2\pi} e^{inx} dx\right) \\
&= \sum_{n=0}^\infty \frac{1}{n!}\int_0^{2\pi} \cos(nx) dx \\
&= \int_0^{2\pi} dx = 2\pi
\end{align}$$

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