 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/26g/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

\frac{1}{u} \int_{- u}^{u} (1 - φ (t)) d t = E [\frac{1}{u} \int_{- u}^{u} (1 - e^{i tX}) d t] .

The odd imaginary part integrates to zero, while

\int_{- u}^{u} e^{i tX} d t = \frac{2 s i n ( u X )}{X} .

Therefore the expression equals

2 E (1 - \frac{s i n ( u X )}{u X}),

with the quotient interpreted continuously at

X = 0

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