 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-2/12/a/solution

Apply the fundamental theorem of calculus separately to the upper and lower limits. No evaluation of the integral is needed:

\frac{\partial I _{n, k}}{\partial x} = cos^{k} x (1 - sin^{3} x)^{n}, \frac{\partial I _{n, k}}{\partial y} = - cos^{k} y (1 - sin^{3} y)^{n} .

The first result has no

y

dependence and the second no

x

dependence, so the mixed partial derivative is

\frac{\partial ^{2} I _{n, k}}{\partial x \partial y} = 0 .

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