Answer
$$\int (\sin2\theta)e^{\sin^2\theta}d\theta=e^{\sin^2\theta}+C$$
Work Step by Step
$$A=\int (\sin2\theta)e^{\sin^2\theta}d\theta$$
We set $u=\sin^2\theta$
Then $$du=(2\sin\theta\cos\theta) d\theta=(\sin2\theta) d\theta$$
Therefore, $$A=\int e^udu= e^u+C$$ $$A=e^{\sin^2\theta}+C$$