 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2013/p/1/c/solution

\int_{a}^{b} p (x) d x = \sum_{j = 0}^{n} f (x_{j}) \int_{a}^{b} ℓ_{j} (x) d x .

w_{j} = \int_{a}^{b} ℓ_{j} (x) d x = \int_{a}^{b} k \neq = j \prod \frac{x - x _{k}}{x _{j} - x _{k}} d x .

The resulting quadrature rule is exact for every polynomial of degree at most

n

, since its interpolant is itself. For a general continuous

f

, replacing

f

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