 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/ii/paper-2/2f/b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Again $f = 0$ . The change of variables

t = x^{2}

gives

0 = \int_{0}^{1} f (x) x^{2 n + 1} d x = \frac{1}{2} \int_{0}^{1} f (t) t^{n} d t (n \geq 0) .

Apply part (i) to the continuous function

h (t) = f (t)

. It vanishes on the whole interval, so does

f

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