Chapter 36

04/19/23By Chtan FYHS-Kulai1

Chapter 36

04/19/23 2By Chtan FYHS-Kulai

Let see the example below :

4

1,12

3

xx

If the 2 fractions are added together, the result :

412

12123

4

1

12

3

xx

xx

xx

04/19/23 3By Chtan FYHS-Kulai

412

13

xx

x

is more complicated than the previous two fractions.

If you want to integrate or expand the fraction, it is much simpler to express it as the sum of the two fractions.

We call these fractions – the partial fractions.

04/19/23 4By Chtan FYHS-Kulai

Expression of a fractional function in partial fractions :

(Rule 1) :

Before a fractional function can be expressed directly in partial fractions the numerator must be of at least one degree less than the denominator.

04/19/23 5By Chtan FYHS-Kulai

e.g. 1

1

323

2

x

x can be expressed in partial fractions.

1

323

3

x

x cannot be expressed directly in partial fractions.

04/19/23 By Chtan FYHS-Kulai 6

1

323

3

x

x can be simplified before it can be expressed as a sum of partial fractions.

1

52

1

3233

3

xx

x

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(Rule 2) :

Corresponding to any linear factor ax+b in the denominator of a rational fraction there is a partial fraction of the form , A is a constant. bax

A

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e.g. 2Express the function in partial fractions. 2121

2

xxx

x

Soln :

21212121

2

x

C

x

B

x

A

xxx

x

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121212122 xxCxxBxxAx

Let x=-2, -4=C(-3)(-3) 9

4C

Let x=1, 2=A(3)(3)9

2A

Let x=-1/2, -1=B(-3/2)(3/2) 9

4B

2

4

12

4

1

2

9

1

2121

2

xxxxxx

x

04/19/23 By Chtan FYHS-Kulai 10

(Rule 3) :

Corresponding to any linear factor ax+b repeated r times in the denominator, there will be r partial fractions of the form

rr

bax

A

bax

A

bax

A

bax

A

,...,, 3

32

21

04/19/23 By Chtan FYHS-Kulai 11

e.g. 3Express as a sum of partial fractions,

11

323

2

xx

x

Soln :

111111

32323

2

x

D

x

C

x

B

x

A

xx

x

04/19/23 By Chtan FYHS-Kulai 12

322 11111132 xDxCxxBxxAx

If x=-1, -1=-8D, D = 1/8

If x=1, -1=2C, C = -1/2

If x=0, -3=A-B+C-D -3=A-B-5/8, A-B=-19/8If x=2, 5=3A+3B+3C+D

5=3A+3B-11/8, A+B=17/8

1

2

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2A=-2/8, A=-1/81+2 :

B=9/4

1

1

1

4-

1

18

1

1-

8

1

11

32323

2

xxxxxx

x

04/19/23 By Chtan FYHS-Kulai 14

(Rule 4) :

Corresponding to any quadratic factor in the denominatorthere will be a partial fraction of the form

cbxax 2

cbxax

BAx

2

04/19/23 By Chtan FYHS-Kulai 15

e.g. 4Express as a sum of partial fractions,

1

24

3

x

x

Soln :

1111

224

3

x

D

x

C

x

BAx

x

x

04/19/23 By Chtan FYHS-Kulai 16

1111112 223 xxDxxCxxBAxx

Put x=1, -1=4D, D=-1/4

Put x=-1, -3=-4C, C=3/4

Put x=0, -2=-B-C+D, -2=-B-3/4-1/4, B=1

Put x=2, 6=(2A+1)(3)+5C+15D 6=3(2A+1), A=1/2

04/19/23 17By Chtan FYHS-Kulai

141-

143

1

121

1

224

3

xxx

x

x

x

1

1

1

3

1

42

4

12 xxx

x

04/19/23 By Chtan FYHS-Kulai 18

Note :

Repeated quadratic factors in the denominator are dealt with in a similar way to repeated linear factors.

22222

1

cbxax

DCx

cbxax

BAx

cbxax

04/19/23 19By Chtan FYHS-Kulai

Ex 16a p. 216 Mathematics 3

Q 3, 4, 5, 6, 9, 11, 12, 13, 15, 17, 20, 23, 27, 30, 32

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The expansion of rational algebraic fractions

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Refer to textbook p.217 example 5 and example 6.

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Ex 16b p. 218 Mathematics 3

Q 3, 5, 9, 11, 13, 17, 20

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The following types of partial fractions will arise :

rrcbxax

BAx

cbxax

BAx

bax

A

bax

A

22

,,,

We can integrate these types of partial fractions.

Beyond the scope of this book .

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e.g. 5Integrate with respectto x.

112 xx

Soln :

111

12

x

C

x

B

x

A

xx

04/19/23 By Chtan FYHS-Kulai 26

11111 xCxxBxxxA

When x=1, 1=2C, C=1/2

When x=-1, 1=2B, B=1/2

When x=0, 1=-A, A=-1

121

121

1-

1

12

xxxxx

1

1

2

1

1

1

2

11-

xxx

04/19/23 By Chtan FYHS-Kulai 27

12

1

12

1

12 x

dx

x

dx

x

dx

xx

dx

cxxx 1ln2

11ln

2

1ln

cx

x

1ln

2

04/19/23 By Chtan FYHS-Kulai 28

Ex 16c p. 220 Mathematics 3

All odd numbers.

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Misc. Ex. p. 221 Mathematics 3

All odd numbers.

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The end