Partial fractions

Warm-up

Three quick questions from earlier lessons.

Discover

$\dfrac{1}{x - 1} - \dfrac{1}{x + 1}$ over a common denominator gives $\dfrac{2}{x^2 - 1}$ . So how can you integrate $\dfrac{1}{x^2 - 1}$ ?

Split it as $\tfrac{1}{2}\left(\tfrac{1}{x - 1} - \tfrac{1}{x + 1}\right)$
Use $\arctan x$

Key idea

Split a rational function into simple fractions, then integrate each to a logarithm.

Partial fractions: if

(x - r)(x - s)

with

r \ne s

, then

\dfrac{p(x)}{(x - r)(x - s)} = \dfrac{A}{x - r} + \dfrac{B}{x - s}

(degree of

p

less than 2). Find

A

by multiplying by

x - r

and setting

x = r

; likewise

B

.

Worked example

$\displaystyle\int \dfrac{5}{(x - 2)(x + 3)}\,dx$ .

A = 1

,

B = -1

x = 2: A = 5/5; x = −3: B = 5/(−5).

\ln|x - 2| - \ln|x + 3| + C

Integrate each.

\ln\left|\dfrac{x - 2}{x + 3}\right| + C

One log.

Your turn, guided

Your turn: $\dfrac{3}{(x - 1)(x + 2)} = \dfrac{A}{x - 1} + \dfrac{B}{x + 2}$ .

A = 1

x = 1: 3/(1 + 2).

Practice ★

$\dfrac{ 8 }{(x - 4)(x - 6)} = \dfrac{A}{x - 4} + \dfrac{B}{x - 6}$ . Find $A$ .
$\dfrac{ 5 }{(x - 1)(x - 3)} = \dfrac{A}{x - 1} + \dfrac{B}{x - 3}$ . Find $A$ .
$\dfrac{ 8 }{(x - 1)(x - 5)} = \dfrac{A}{x - 1} + \dfrac{B}{x - 5}$ . Find $A$ .

Watch out!

Practice ★★

Evaluate $\displaystyle\int_1^2 \dfrac{dx}{x(x + 3)}$ .
Evaluate $\displaystyle\int_1^2 \dfrac{dx}{x(x + 4)}$ .

Evaluate $\displaystyle\int_{ 5 }^{ 6 } \dfrac{dx}{x^2 - 1}$ .

Exit ticket

Three questions to check you have got it.

$\dfrac{ 8 }{(x - 4)(x - 6)} = \dfrac{A}{x - 4} + \dfrac{B}{x - 6}$ . Find $A$ .
$\dfrac{ 5 }{(x - 1)(x - 3)} = \dfrac{A}{x - 1} + \dfrac{B}{x - 3}$ . Find $A$ .
$\dfrac{ 8 }{(x - 1)(x - 5)} = \dfrac{A}{x - 1} + \dfrac{B}{x - 5}$ . Find $A$ .

Summary card

Partial fractions

Factor the denominator.
Find $A$ , $B$ by cover-up.
Integrate: $\displaystyle\int \dfrac{A}{x - r}\,dx = A\ln|x - r|$ .

Watch out: check by recombining the fractions.

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