 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-2/11/a/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

\frac{3 x ^{2} + 5 x + 1}{x ( x + 1 ) ( x + 2 )} = \frac{1}{2 x} + \frac{1}{x + 1} + \frac{3}{2 ( x + 2 )} .

Integrating between one and two gives

\frac{1}{2} ln 2 + ln \frac{3}{2} + \frac{3}{2} ln \frac{4}{3} = \frac{1}{2} ln \frac{32}{3} .

All logarithm arguments are positive on this integration interval.

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