 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/3/c/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

The Gaussian decay makes the integral absolutely convergent, so odd symmetry gives

\int_{- \infty}^{\infty} f (x) d x = 0

. On the positive half-line substitute

u = 3 x^{2} /2

, so

x^{3} d x = 2 u d u /9

. Thus

\int_{0}^{\infty} f (x) d x = \frac{2}{27} \int_{0}^{\infty} u e^{- u} d u = \frac{2}{27} .

The last integral is

Γ (2) = 1

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