chapter 36
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Chapter 36. Partial fractions & their application. Let see the example below :. If the 2 fractions are added together, the result :. is more complicated than the previous two fractions. - PowerPoint PPT PresentationTRANSCRIPT
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
04/19/23 By Chtan FYHS-Kulai 8
e.g. 2Express the function in partial fractions. 2121
2
xxx
x
Soln :
21212121
2
x
C
x
B
x
A
xxx
x
04/19/23 By Chtan FYHS-Kulai 9
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
04/19/23 By Chtan FYHS-Kulai 13
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
04/19/23 By Chtan FYHS-Kulai 20
The expansion of rational algebraic fractions
04/19/23 By Chtan FYHS-Kulai 21
Refer to textbook p.217 example 5 and example 6.
04/19/23 By Chtan FYHS-Kulai 22
Ex 16b p. 218 Mathematics 3
Q 3, 5, 9, 11, 13, 17, 20
04/19/23 By Chtan FYHS-Kulai 23
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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 .
04/19/23 By Chtan FYHS-Kulai 25
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.
04/19/23 By Chtan FYHS-Kulai 29
Misc. Ex. p. 221 Mathematics 3
All odd numbers.
04/19/23 By Chtan FYHS-Kulai 30
The end