integral

$\int_{0}^{\infty} \frac{e^{\cos x} \sin (\sin x)}{x} \ dx = \frac{\pi}{2} (e-1)$

It's tempting to say that $\int_{0}^{\infty} \frac{e^{\cos x} \sin (\sin x)}{x} \ dx = \text{Im} \int_{0}^{\infty} \frac{e^{e^{ix}}}{x} \ dx$ .

But $\int_{0}^{\infty} \frac{e^{e^{ix}}}{x} \ dx$ diverges.

EDIT: A principal value approach works.