Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 20 / a

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Solution

The Leibniz integral rule combines contributions from both moving endpoints with differentiation under the integral sign:

\frac{d}{d x} \int_{a (x)}^{b (x)} f (x, t) d t = f (x, b (x)) b^{'} (x) - f (x, a (x)) a^{'} (x) + \int_{a (x)}^{b (x)} \frac{\partial f}{\partial x} (x, t) d t .

The partial derivative in the last integral holds

t

fixed. Suitable regularity and convergence are assumed as stipulated.

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