Elementary methods to evaluate $\int_{0}^{2\pi}\exp(\sin \theta)\cos(\cos \theta)\mathrm d\theta$ I have seen similar integrals being attacked by complex analysis. I am wondering whether there are elementary methods to solve such integrals.

$$\int_{0}^{2\pi}\exp(\sin\theta)\cos(\cos\theta)\mathrm d\theta$$

Any hints are appreciated. Thanks.
 A: This is inspired by a deleted solution of a similar question (Calculating the following integral using complex analysis: $\int_{0}^{\pi}e^{a\cos(\theta)}\cos(a\sin(\theta))\, d\theta$), containing no details, though:
Let $$I(a)=\int^{2\pi}_0 e^{a \sin \theta} \cos(a \cos \theta)\,d\theta.$$ Then,
$$\frac{d}{d a}I(a) = \int^{2\pi}_0 e^{a \sin \theta} (\sin \theta\cos(a \cos \theta) - \cos \theta\sin(a \cos \theta))\,d\theta.\tag{1}$$ Since
$$\frac{d}{d\theta}(e^{a \sin \theta} \sin(a \cos \theta)) = -a e^{a \sin \theta} (\sin \theta\cos(a \cos \theta) - \cos \theta\sin(a \cos \theta)),$$
the RHS of (1) is $0$ for $a>0$, i.e. $I(a)$ is a constant, implying
$$I(a)=\lim_{a\to0^+}I(a)=2\pi.\tag{2}$$
Differentiation under the integral sign and the limit in (2) can be justified using the standard ("elementary") methods of real anaysis, like dominated convergence theorem etc.
I guess the solution was deleted because the question explicitly asked for an answer using complex analysis. Let's face it: to distinguish between "elementary" and complex analysis was dubious (and mere tradition) already 100 years ago, it's absolutely ludicrous in the 21st century.
