Answer
$$-\frac{1}{2}e^{\cos^2\theta +1}+c.$$
Work Step by Step
Let $u=\cos^2\theta +1$, then $du=-2\cos \theta \sin \theta d\theta$, hence we have
$$\int \cos \theta \sin \theta e^{\cos^2\theta +1}d\theta=-\frac{1}{2}\int e^udu\\
=-\frac{1}{2}e^u+c=-\frac{1}{2}e^{\cos^2\theta +1}+c.$$
