Find the leading order asymptotics of $$I_n(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} e^{x\cos(\theta)}\cos(n\theta)d\theta$$ as $x \to +\infty$.
Here's my work:
I am using the Laplace method: $\cos(\theta)$ has a (unique) max at $c=0\in(-\pi,\pi)$, $-\sin(0)=0$, and $-\cos(0)<0$. $\cos(\theta)$ is smooth and $\cos(n\theta)$ is continuous.
So then, we use $c=0$ since the main contribution comes from $c$ to approximate i.e. replace $\cos(n\theta)$ with $\cos(0)$ and $\cos(\theta)$ with the expansion of $\cos(0)-\sin(0)(\theta-0)-\frac{\cos(0)(\theta-0)^2}{2}$.
Then, $$I_n(x)\sim \frac{1}{2\pi}\cos(0)e^{x\cos(0)}\sqrt{\frac{-2\pi}{-x\cos(0)}}=\frac{1}{2\pi}e^x\sqrt{\frac{2\pi}{x}}$$ as $x \to +\infty$.
Am I right or on the right track? I don't have much experience with asymptotic expansion of integrals so I am having trouble with the reading on what I can find (e.g. Laplace Method, etc.) I also tried to follow Leading terms in asymptotic expansion of modified bessel function of the first kind due to the similiarity of the problem, but wasn't able to arrive at a solid understanding.