$$\int_0^{2\pi} \exp(int) \exp(\cos(t))\; dt = \int_{-\pi}^{\pi} \cos(n t) \exp(\cos(t))\; dt = 2 \pi I_n(1)$$ where $I_n$ is a modified Bessel function of the first kind.

$$\int_0^{2\pi} \exp(int) \exp(\cos(t))\; dt = \int_{-\pi}^{\pi} \cos(n t) \exp(\cos(t))\; dt = 2 \pi I_n(1)$$ where $I_n$ is a modified Bessel function of the first kind and thus $$I_n(1)=\frac12\sum_{k\geq0}\frac1{4^kk!(n+k)!}.$$