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Compute the definite integral: $\int_{0}^{2\pi} e^{\cos \theta} \cos(\sin\theta) \, d\theta.$

Hint: Consider the function $f(t) = \int_{0}^{2\pi} e^{t \cos\theta} \cos(t\sin\theta) \, d\theta$

and use differentiation under the integral sign.

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