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 $\displaystyle\int_{0}^{1} \dfrac{x + 5}{(x + 1)(x + 3)}\,dx = \ln 3$ . (6 marks)
Level 4 · 6 marksTry it
2
$f(x) = \dfrac{11x - 1}{(x - 1)(2x + 3)}$ , $\quad x > 1$ .
(a) Express $f(x)$ in partial fractions. (3 marks)
(b) Hence show that

$\displaystyle\int_2^6 f(x)\; dx = 2\ln 5 + \frac{7}{2}\ln\frac{15}{7}$ . (5 marks)
Level 4 · 8 marksTry it
3
In this question you must show detailed reasoning.

Show that

$\displaystyle\int_1^4 \frac{2x+1}{x(x+2)}\,\mathrm{d}x = a\ln 2$ ,

where $a$ is a rational number to be determined. (7 marks)
Level 4 · 7 marksTry it