 Solution

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

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

Write

sinh (x Φ) = (e^{x Φ} - e^{- x Φ}) /2

Φ

is the nonstationary endpoint

t = 1

, where

Φ (1) = 1, Φ^{'} (1) = 6.

Endpoint Laplace's method therefore gives

\frac{1}{2} \int_{0}^{1} e^{x Φ (t)} d t \sim \frac{e ^{x}}{12 x} .

The second exponential has its maximum at

t = 1/2

and is only of order

x^{- 1/2} e^{x /8}

, exponentially smaller. Hence

J (x) \sim \frac{e ^{x}}{12 x} .

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