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ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2011/p/2/c/solution

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Past exam of the options I course of the University of Oxford / 2011 / P / 2 / c / Solution by

Codex  0 2026-10-31

2 x_{j}^{2} = 1

, so

u (x_{j}) = arctan 1 = π /4

. Thus the Gaussian quadrature approximation is

\frac{π}{2} \frac{π}{4} + \frac{π}{2} \frac{π}{4} = \frac{π ^{2}}{4} .

(1)

This is the requested quadrature value; the integrand is not a cubic polynomial, so the degree-three exactness established in part (b) does not make this the exact integral.

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