Integration usingpartial fractions 413
=
1
2 a
[ln(x−a)−ln(x+a)]+c
=
1
2 a
ln
(
x−a
x+a
)
+c
Problem 10. Evaluate
∫ 4
3
3
(x^2 − 4 )
dx,
correct to 3 significant figures.
From Problem 9,
∫ 4
3
3
(x^2 − 4 )
dx= 3
[
1
2 ( 2 )
ln
(
x− 2
x+ 2
)] 4
3
=
3
4
[
ln
2
6
−ln
1
5
]
=
3
4
ln
5
3
= 0. 383 ,correct to 3
significant figures.
Problem 11. Determine
∫
1
(a^2 −x^2 )
dx.
Using partial fractions, let
1
(a^2 −x^2 )
≡
1
(a−x)(a+x)
≡
A
(a−x)
+
B
(a+x)
≡
A(a+x)+B(a−x)
(a−x)(a+x)
Then 1≡A(a+x)+B(a−x)
Letx=athenA=
1
2 a
.Letx=−athenB=
1
2 a
Hence
∫
1
(a^2 −x^2 )
dx
=
∫
1
2 a
[
1
(a−x)
+
1
(a+x)
]
dx
=
1
2 a
[−ln(a−x)+ln(a+x)]+c
=
1
2 a
ln
(
a+x
a−x
)
+c
Problem 12. Evaluate
∫ 2
0
5
( 9 −x^2 )
dx,
correct to 4 decimal places.
From Problem 11,
∫ 2
0
5
( 9 −x^2 )
dx= 5
[
1
2 ( 3 )
ln
(
3 +x
3 −x
)] 2
0
=
5
6
[
ln
5
1
−ln1
]
= 1. 3412 ,correct to 4 decimal places.
Now try the following exercise
Exercise 164 Further problems on
integration using partial fractions with
quadratic factors
- Determine
∫
x^2 −x− 13
(x^2 + 7 )(x− 2 )
dx.
⎡
⎣ln(x
(^2) + 7 )+√^3
7
tan−^1
x
√
7
−ln(x− 2 )+c
⎤
⎦
In Problems 2 to 4, evaluate the definite integrals
correct to 4 significant figures.
2.
∫ 6
5
6 x− 5
(x− 4 )(x^2 + 3 )
dx [0.5880]
3.
∫ 2
1
4
( 16 −x^2 )
dx [0.2939]
4.
∫ 5
4
2
(x^2 − 9 )
dx [0.1865]
- Show that
∫ 2
1
(
2 +θ+ 6 θ^2 − 2 θ^3
θ^2 (θ^2 + 1 )
)
dθ
=1.606, correct to 4 significant figures.