 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-3/19c/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Let

x_{1}, \dots, x_{N}

be the distinct real roots of

He_{N}

\int_{- \infty}^{\infty} f (x) e^{- x^{2} /2} d x \approx \sum_{j = 1}^{N} w_{j} f (x_{j}),

where

w_{j}

is the weighted integral of the

j

w_{j} = \frac{N ! 2 π}{N ^{2} He _{N - 1} ( x _{j} ) ^{2}} > 0.

He_{N}

makes the rule exact for every polynomial of degree at most

2 N - 1 .

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