= Solution
Use the given <Cartesian coordinates> and integrate with respect to $y$ first. Since $\cos y\,e^{\sin y}$ is the <derivative> of $e^{\sin y}$, the <double integral> becomes
$$
I=\int_0^\pi\cos x\left[ e^{\sin y}\right]_{y=0}^{y=x}\,dx
=\int_0^\pi\cos x(e^{\sin x}-1)\,dx.
$$
A direct <antiderivative> avoids any issue with the nonmonotonicity of $\sin x$ on this interval:
$$
\boxed{I=\left[e^{\sin x}-\sin x\right]_0^\pi=0}.
$$
The sign changes in $\cos x$ make this cancellation possible even though $e^{\sin y}$ is positive.
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