 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/13/d/solution

The sum is finite, so differentiation passes through it without a convergence issue. Each lower limit

n

is constant with respect to

x

, and the fundamental theorem of calculus gives the same upper-end integrand. Hence the derivative is

sin (x^{2} + x^{6}) \sum_{n = 0}^{N} (n N) .

1 + 1

gives the sum

2^{N}

. Therefore

\frac{d}{d x} [n = 0 \sum N (n N) \int_{n}^{x} sin (y^{2} + y^{6}) d y] = 2^{N} sin (x^{2} + x^{6}) .

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