Evaluate $\int_0^{\pi} e^{a\cos(t)}\cos(a\sin t)dt$ What is the value of
$$\int_0^{\pi} e^{a\cos(t)}\cos(a\sin t)dt?$$  
 A: By exploiting $\cos\theta=\frac{1}{2}\left(e^{i\theta}+e^{-i\theta}\right)$ and the Taylor series of the exponential function we have:
$$\begin{eqnarray*}I(a)&=&\int_{0}^{\pi}e^{a\cos t}\cos(a\sin t)\,dt\\&=&\frac{1}{2}\int_{0}^{\pi}\exp\left(a e^{it}\right)+\exp(ae^{-it})\,dt\\&=&\frac{1}{2}\sum_{n\geq 0}\int_{0}^{\pi}\frac{a^n(e^{nit}+e^{-nit})}{n!}\,dt\end{eqnarray*} $$ 
but the innermost integral always vanishes, unless $n=0$. It follows that $I(a)$ is constant, and
$$ I(0) = \pi $$
is trivial.
A: The integral may be expressed at
$$\frac12 \operatorname{Re} \int_0^{2 \pi} dt \, e^{a e^{i t}} $$
The integral may be written in complex form as, letting $z=e^{i t}$:
$$-i \oint_{|z|=1} dz \frac{e^{a z}}{z} $$
which by Cauchy's integral theorem is simply $2 \pi$.  Thus, the sought-after integral is $\pi$.
A: Let us calculate the derivative of the integral $I(a)$ with respect to $a$
$$
I'(a)=\int_0^{\pi}e^{a \cos(t)}\cos(t)\cos(a\sin(t))-\int_0^{\pi}e^{a \cos(t)}\sin(t)\sin(a\sin(t))
$$
applying integration by parts to the second integral with $u'=e^{a \cos(t)}\sin(t)$ and $v=\sin(a\sin(t))$
we get 
$$
I'(a)=0
$$
so 
$$
I(a)=const
$$
but $I(0)=\pi$ and we end up with

$$
I(a)=\pi
$$

A: I got a really strange/amusing/interesting response, with Wolfram Mathematica.
In few words, as long as $-24 \leq a \leq 24.60$ the value is:
$$\int_0^{\pi}\ e^{a \cos(t)}\cos(a \sin(t)) \text{d}t = \pi$$
