 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/20/a/solution

Under the regularity and integrability conditions for the Leibniz integral rule,

\frac{d}{d θ} \int_{a (θ)}^{b (θ)} f (x, θ) d x = f (b (θ), θ) b^{'} (θ) - f (a (θ), θ) a^{'} (θ) + \int_{a (θ)}^{b (θ)} \frac{\partial f}{\partial θ} (x, θ) d x .

The first two terms account for the moving endpoints; the last accounts for the integrand's own parameter dependence.

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