Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 15 / a / 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 becomesA direct antiderivative avoids any issue with the nonmonotonicity of $\sin x$ on this interval:The sign changes in $\cos x$ make this cancellation possible even though $e^{\sin y}$ is positive.