Integrate $\int_0^{2\pi} e^{\cos\theta} \sin(\sin\theta-n\theta)d\theta$

Question

How to integrate

1. $\int_0^{2\pi} e^{\cos\theta} \sin(\sin\theta-n\theta)d\theta$
2. $\int_0^{2\pi} e^{-\cos\theta} \cos(\sin\theta+n\theta)d\theta$

Answer 1

The first integral is$$\Im\int_0^{2\pi}e^{\cos\theta+i(\sin\theta-n\theta)}d\theta=\Im\int_0^{2\pi}e^{e^{i\theta}-in\theta}d\theta.$$Expanding $e^{e^{i\theta}}$ as a power series in $e^{i\theta}$ and using $\int_0^{2\pi}e^{ik\theta}=2\pi\delta_{k0}$ for $k\in\Bbb Z$, this is$$\Im\frac{2\pi}{k!}=0.$$Similarly, the second integral is$$\Re\int_0^{2\pi}e^{-e^{-i\theta}+in\theta}d\theta=\frac{(-1)^n2\pi}{n!}.$$