I need to evaluate the following definite integral:
$$J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$$
I have attempted basic variable substitution and expanding the cosine term, but I have not been able to find an indefinite integral. I believe the best strategy would be to use contour integration, but I am not sure on what contour I can use and how I can proceed with such a method.
Rewrite the integral as $$I = \text{Real}\left(\int_{0}^{2\pi} e^{\cos(x)-i \sin(x) +2ix} dx \right)$$ Hence, we have $$\int_{0}^{2\pi} e^{\cos(x)-i \sin(x) +2ix} dx = \int_{0}^{2\pi} e^{2ix}e^{e^{-ix}}dx = \sum_{k=0}^{\infty} \int_0^{2\pi}e^{2ix}\dfrac{e^{-ikx}}{k!}dx = \pi$$ where only the $k=2$ term survives in the last summation, since $\int_0^{2\pi}e^{imx}dx = 2\pi \delta_{m,0}$. Hence, $$I = \pi$$