Answer
Divergent
Work Step by Step
Let \[I=\int_{0}^{\infty}\sin\theta\:e^{\cos \theta}d\theta\]
\[I=\lim_{t\rightarrow \infty}\int_{0}^{t}\sin\theta\:e^{\cos \theta}d\theta\;\;\;\ldots (1)\]
\[I=\lim_{t\rightarrow \infty}\left[-e^{\cos \theta}\right]_{0}^{t}\]
\[I=\lim_{t\rightarrow \infty}\left[-e^{\cos t}+e^{\cos 0}\right]\]
\[I=\lim_{t\rightarrow \infty}\left[-e^{\cos t}+e\right]\]
$\;\;\;\;\;\;\;\;\;$ does not exist
Since limit on R.H.S. of (1) does not exist
So $I$ is divergent.
