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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/7/b/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 7 / b / Solution by

Codex  0 2026-10-18

Additivity of the definite integral combines the adjacent intervals into

[0, n]

. For an integer

n \geq 1

, the power rule gives

\sum_{j = 0}^{n - 1} \int_{j}^{j + 1} (x - 2)^{n} d x = \int_{0}^{n} (x - 2)^{n} d x = \frac{( n - 2 ) ^{n + 1} - ( - 2 ) ^{n + 1}}{n + 1} .

(1)

For

n = 0

, interpret the indexed sum as an empty sum, equal to zero; the displayed expression also gives zero. This convention resolves the otherwise degenerate endpoint of the stated nonnegative-integer range.

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