A-level · Integration

Integrating with partial fractionsExam-style questions

Integrating a rational function by splitting it into partial fractions, so each piece integrates to a logarithm. When a fraction needs splitting; finding the numerators; integrating each piece, with the 1/a for a factor ax + b; definite integrals tidied with the laws of logs; dividing first when the fraction is improper; and repeated factors, which give a power rather than a log.

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18 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
Show that $\displaystyle\int_{2}^{3} \dfrac{2}{x(x - 1)}\,dx = 2\ln\dfrac{4}{3}$ . (5 marks)
Level 5 · 5 marksTry it
2
The diagram shows part of the curve $y = \dfrac{2x^2 + 6x + 7}{(x + 1)(x + 2)}$ . The shaded region is bounded by the curve, the $x$ -axis and the lines $x = 1$ and $x = 4$ .

Find the exact area of the shaded region. Give your answer in the form $p + q\ln r$ , where $p$ and $q$ are positive integers and $r$ is a rational number greater than $1$ .

(10 marks)
Level 5 · 10 marksTry it