Solution
= Solution
Apply the <fundamental theorem of calculus> separately to the upper and lower limits. No evaluation of the <integral> is needed:
$$
\boxed{\frac{\partial I_{n,k}}{\partial x}=\cos^kx(1-\sin^3x)^n,\qquad
\frac{\partial I_{n,k}}{\partial y}=-\cos^ky(1-\sin^3y)^n}.
$$
The first result has no $y$ dependence and the second no $x$ dependence, so the mixed <partial derivative> is
$$
\boxed{\frac{\partial^2I_{n,k}}{\partial x\partial y}=0}.
$$