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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 16 / a 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 15 / b 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 15 / a 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 14 / b 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 14 / a 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 13 / b 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 13 / a 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 12 / d 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 12 / c 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 12 / b 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 12 / a 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 11 / c 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 11 / b 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 11 / a 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 10 / b 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 10 / a 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 1 / b 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 1 / a 2026-10-18

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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 2026-10-18

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The partial fraction decomposition is

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

(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.

(2)

Hence

n = 2

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 8 / Solution 2026-10-18

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Apply implicit differentiation, remembering that

y

depends on

x

1 + y^{'} + e^{x} + e^{y} y^{'} = 0.

(1)

Since

1 + e^{y} > 0

, division is always legitimate. Therefore

\frac{d y}{d x} = - \frac{1 + e ^{x}}{1 + e ^{y}} .

(2)

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