 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/9/solution

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

Taking antiderivatives and using the positive endpoints gives

\int_{2}^{3} \frac{2 x + 1}{x ( x + 1 )} d x = [ln x + ln (x + 1)]_{2}^{3} = ln \frac{3 \cdot 4}{2 \cdot 3} = ln 2.

Hence

n = 2

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