A-level · Algebra & functions

Partial fractionsExam-style questions

Splitting one fraction into a sum of simpler ones. Partial fractions as adding fractions run backwards, and why the pieces are worth having; distinct linear factors, found by substituting the values that make brackets vanish; factorising the denominator first, including factors like (2x + 1); the extra piece a repeated factor needs, and equating coefficients for it; and dividing out a top-heavy fraction first.

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

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
(a) Express $\dfrac{3x + 4}{(x + 1)(x + 2)}$ in partial fractions. (4 marks)
(b) Hence show that $\displaystyle\int_{0}^{1} \dfrac{3x + 4}{(x + 1)(x + 2)} \, dx = \ln \dfrac{9}{2}$ . (3 marks)
Level 5 · 7 marksTry it
2
$\mathrm{f}(x) = \dfrac{9}{(x - 1)(x + 2)^2}$
(a) Express $\mathrm{f}(x)$ in partial fractions. (5 marks)
(b) Hence find $\displaystyle\int \mathrm{f}(x)\;\mathrm{d}x$ . (4 marks)
(c) Hence show that

$\int_2^5 \mathrm{f}(x)\;\mathrm{d}x = \ln\dfrac{16}{7} - \dfrac{9}{28}$

(3 marks)
Level 5 · 12 marksTry it
3
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Given that

$\dfrac{3x^2 - 6x + 9}{(x + 1)(x - 2)^2} \equiv \dfrac{A}{x + 1} + \dfrac{B}{x - 2} + \dfrac{C}{(x - 2)^2}$
(a) find the values of the constants $A$ , $B$ and $C$ .

(4 marks)
(b) Use your answer to part (a) to find the exact value of

$\int_3^5 \dfrac{3x^2 - 6x + 9}{(x + 1)(x - 2)^2}\,dx$

writing your answer in the form $p\ln q + r$ , where $p$ , $q$ and $r$ are rational numbers.

(6 marks)
Level 5 · 10 marksTry it
4
$f(x) = \dfrac{2}{(1 + 2x)(1 - x)}$ .
(a) Express $f(x)$ in partial fractions. (3 marks)
(b) Hence show that, for small $x$ , $f(x) \approx 2 - 2x + 6x^2$ . (6 marks)
(c) State the range of values of $x$ for which the full expansion is valid, justifying your answer. (3 marks)
Level 5 · 12 marksTry it
5
(a) Express $\dfrac{3x^{2}+6}{(x+2)(x-1)^{2}}$ in partial fractions. (5 marks)
(b) Hence show that $\displaystyle\int_{2}^{4}\dfrac{3x^{2}+6}{(x+2)(x-1)^{2}}\,dx = \ln\dfrac{27}{4} + 2$ . (3 marks)
Level 5 · 8 marksTry it