Answer
$${\text{Diverges}}$$
Work Step by Step
$$\eqalign{
& \int_0^\infty {\sin \theta {e^{\cos \theta }}d\theta } \cr
& {\text{Use the definition of improper integrals}} \cr
& \int_0^\infty {\sin \theta {e^{\cos \theta }}d\theta } = \mathop {\lim }\limits_{b \to \infty } \int_0^b {\sin \theta {e^{\cos \theta }}d\theta } \cr
& {\text{Integrating}} \cr
& = \mathop {\lim }\limits_{b \to \infty } \left[ { - {e^{\cos \theta }}} \right]_0^b \cr
& = - \mathop {\lim }\limits_{b \to \infty } \left[ {{e^{\cos b}} - {e^{\cos 0}}} \right] \cr
& = - \mathop {\lim }\limits_{b \to \infty } \left[ {{e^{\cos b}} - e} \right] \cr
& {\text{Evaluate the limit when }}b \to \infty \cr
& = - \left[ {{e^{\mathop {\lim }\limits_{b \to \infty } \left[ {\cos b} \right]}} - e} \right] \cr
& *\cos \left( \infty \right){\text{ oscillates between }}\left[ { - 1,1} \right] \cr
& {\text{The limit does not exist, then the integral diverges}}{\text{.}} \cr} $$
