= Solution
The <phase function> $\cos t$ attains its global maximum $1$ on $[0,3\pi]$ at the stationary endpoint $t=0$ and at the interior point $t=2\pi$. At either point,
$$
\cos(t_0+u)=1-\frac{u^2}{2}+O(u^4).
$$
By <Laplace's method at a nondegenerate stationary endpoint>, the endpoint contributes
$$
e^x\sqrt{\frac{\pi}{2x}}.
$$
By <Laplace's method>, the interior maximum contributes
$$
e^{x+2\pi}\sqrt{\frac{2\pi}{x}}.
$$
The remaining parts of the interval are exponentially smaller. Summing the two contributions according to <Laplace's method with multiple global maxima> gives
$$
\boxed{
I(x)\sim e^x\sqrt{\frac{\pi}{2x}}
\left(1+2e^{2\pi}\right)}.
$$
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