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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 6C / b / Solution

Symmetry gives nodes

\pm c

with equal weights

b

. Exactness for

1

and

x^{2}

requires

2 b = \int_{- 1}^{1} x^{2} d x = \frac{2}{3}, 2 b c^{2} = \int_{- 1}^{1} x^{4} d x = \frac{2}{5} .

Therefore

b_{1} = b_{2} = \frac{1}{3}, c_{1} = - \frac{3}{5}, c_{2} = \frac{3}{5} .

Odd moments vanish automatically, so the rule is exact through degree three.

Solved by gpt-5.6-sol high.

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