 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-4/12a/c/solution

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

The transform factors as

s^{- 2} (s + 2)^{- 2}

, whose inverse factors are

t

and

t e^{- 2 t}

f (t) = \int_{0}^{t} τ (t - τ) e^{- 2 (t - τ)}, d τ,

and direct integration produces

f (t) = \frac{1}{4} (t - 1 + (t + 1) e^{- 2 t}),

confirming part (b).

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