412 Higher Engineering Mathematics
Now try the following exercise
Exercise 163 Further problems on
integration using partial fractions with
repeated linear factors
In Problems 1 and 2, integrate with respect
tox.
1.
∫
4 x− 3
(x+ 1 )^2
dx
[
4ln(x+ 1 )+
7
(x+ 1 )
+c
]
2.
∫
5 x^2 − 30 x+ 44
(x− 2 )^3
dx
⎡
⎢
⎢
⎣
5ln(x− 2 )+
10
(x− 2 )
−
2
(x− 2 )^2
+c
⎤
⎥
⎥
⎦
In Problems 3 and 4, evaluate the definite integrals
correct to 4 significant figures.
3.
∫ 2
1
x^2 + 7 x+ 3
x^2 (x+ 3 )
[1.663]
4.
∫ 7
6
18 + 21 x−x^2
(x− 5 )(x+ 2 )^2
dx [1.089]
- Show that
∫ 1
0
(
4 t^2 + 9 t+ 8
(t+ 2 )(t+ 1 )^2
)
dt= 2. 546 ,
correct to 4 significant figures.
41.4 Worked problemson
integration using partial
fractions with quadratic factors
Problem 8. Find
∫
3 + 6 x+ 4 x^2 − 2 x^3
x^2 (x^2 + 3 )
dx.
It was shown in Problem 9, page 18:
3 + 6 x+ 4 x^2 − 2 x^3
x^2 (x^2 + 3 )
≡
2
x
+
1
x^2
+
3 − 4 x
(x^2 + 3 )
Thus
∫
3 + 6 x+ 4 x^2 − 2 x^3
x^2 (x^2 + 3 )
dx
≡
∫(
2
x
+
1
x^2
+
( 3 − 4 x)
(x^2 + 3 )
)
dx
=
∫{
2
x
+
1
x^2
+
3
(x^2 + 3 )
−
4 x
(x^2 + 3 )
}
dx
∫
3
(x^2 + 3 )
dx= 3
∫
1
x^2 +(
√
3 )^2
dx
=
3
√
3
tan−^1
x
√
3
, from 12, Table 40.1, page 399.
∫
4 x
x^2 + 3
dxis determined using the algebraic substi-
tutionu=(x^2 + 3 ).
Hence
∫ {
2
x
+
1
x^2
+
3
(x^2 + 3 )
−
4 x
(x^2 + 3 )
}
dx
=2lnx−
1
x
+
3
√
3
tan−^1
x
√
3
−2ln(x^2 + 3 )+c
=ln
(
x
x^2 + 3
) 2
−
1
x
+
√
3tan−^1
x
√
3
+c
Problem 9. Determine
∫
1
(x^2 −a^2 )
dx.
Let
1
(x^2 −a^2 )
≡
A
(x−a)
+
B
(x+a)
≡
A(x+a)+B(x−a)
(x+a)(x−a)
Equating the numerators gives:
1 ≡A(x+a)+B(x−a)
Let x=a,thenA=
1
2 a
,andletx=−a,then
B=−
1
2 a
Hence
∫
1
(x^2 −a^2 )
dx
≡
∫
1
2 a
[
1
(x−a)
−
1
(x+a)
]
dx