 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-2/32a/b/solution

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

Let

Φ (t) = 2 t^{4} - t^{2}

[0, 1]

are the endpoint

0

and the interior point

1/2

, with

Φ (0) = 0, q u a d Φ^{''} (0) = - 2, Φ (1/2) = - \frac{1}{8}, q u a d Φ^{''} (1/2) = 4.

The endpoint stationary-phase contribution to

\int_{0}^{1} e^{i x Φ (t)} d t

\frac{π}{2 x} e^{- iπ /4},

and the interior contribution is

\frac{π}{2 x} e^{i (π /4 - x /8)} .

The nonstationary endpoint contributes only

O (x^{- 1})

I (x) \sim \frac{π}{2 x} [sin (\frac{π}{4} - \frac{x}{8}) - \frac{1}{2}] .

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