 Solution

ID: 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/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 .

t

fixed. Suitable regularity and convergence are assumed as stipulated.

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