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

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

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

EDIT: A principal value approach works.