Answer
Diverges
Work Step by Step
$$\eqalign{
& \int_0^\infty {\sin \theta {e^{\cos \theta }}} d\theta \cr
& {\text{Using the definition of improper integrals }} \cr
& \underbrace {\int_a^\infty {f\left( x \right)dx = \mathop {\lim }\limits_{t \to \infty } } \int_a^t {f\left( x \right)} dx}_ \Downarrow \cr
& \int_0^\infty {\sin \theta {e^{\cos \theta }}} d\theta = \mathop {\lim }\limits_{t \to \infty } \int_0^t {\sin \theta {e^{\cos \theta }}} d\theta \cr
& = - \mathop {\lim }\limits_{t \to \infty } \int_0^t {{e^{\cos \theta }}} \left( { - \sin \theta } \right)d\theta \cr
& {\text{Integrating}} \cr
& = - \mathop {\lim }\limits_{t \to \infty } \left[ {{e^{\cos \theta }}} \right]_0^t \cr
& = - \mathop {\lim }\limits_{t \to \infty } \left[ {{e^{\cos t}} - {e^{\cos 0}}} \right] \cr
& = {e^{\left( {\mathop {\lim }\limits_{t \to \infty } \cos t} \right)}} - e \cr
& {\text{Evaluate the limit when }}t \to \infty \cr
& \mathop {\lim }\limits_{t \to \infty } \cos t{\text{ oscillates between }} \pm {\text{1}}{\text{, the limit does not exist}}{\text{.}} \cr
& {\text{Therefore}}{\text{, the integral diverges.}} \cr} $$
