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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/20/b/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 20 / b / Solution by

Codex  0 2026-10-18

Apply the Leibniz integral rule, including both moving boundaries and the explicit parameter in the integrand. The upper bound has derivative

- sin x

, the lower bound has derivative

cos x

, and the integrand has partial derivative

- t^{4} e^{- x t^{4}}

with respect to

x

. Therefore

f^{'} (x) = - sin x e^{- x c o s^{4} x} - cos x e^{- x s i n^{4} x} - \int_{s i n x}^{c o s x} t^{4} e^{- x t^{4}} d t .

(1)

The formula remains valid when the bounds coincide or occur in reversed order, by the oriented definite integral convention. The integrand and its partial derivative are continuous on every finite integration interval.

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