Evaluating: $\int^\pi_0 \cos(a(\sin(x))e^{a\cos(x)} dx$

Question

How can I evaluate the following integral?

I don't know how to start...

$$\int^\pi_0 \cos(a(\sin(x))e^{a\cos(x)} dx$$ Where $a \ne0, a\in R $

Answer 1

(Here is a sketch not a full solution)

Define $f(a)=\cos(a\sin x) e^{a \cos x}$. Take Laplace transform to obtain $$\mathcal{L}\{f(a)\}=\frac{s-\cos x}{1+s^2-2s \cos x}$$ hence \begin{align} \int_0^{\pi}\cos(a\sin x) e^{a \cos x}dx&=\mathcal{L^{-1}}\{\int_0^{\pi}\frac{s-\cos x}{1+s^2-2s \cos x}dx\}\\ &=\mathcal{L^{-1}}\{\frac{\pi}{s}\}\\ &=\pi \end{align}

Answer 2

$$\int_{0}^{\pi}\cos(a\sin x)\,e^{a\cos x}\,dx = \text{Re}\int_{0}^{\pi}\exp\left(a e^{ix}\right)\,dx $$ Now there is an easy way to evaluate the last integral, that is to exploit the Taylor series of the exponential function and the fact that for any $n\in\mathbb{N}_{> 0}$ we have:

$$ \int_{0}^{\pi}\left(a e^{ix}\right)^n\,dx = \frac{i a^n}{n}(1-(-1)^n)$$ so the only term that survives to $\text{Re}\int$ is the one associated with $n=0$ and: $$\int_{0}^{\pi}\cos(a\sin x)\,e^{a\cos x}\,dx = \color{red}{\pi}.$$