Chapter 36
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Transcript of 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
04/19/23 By Chtan FYHS-Kulai 7
(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 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 24
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