Higher Engineering Mathematics, Sixth Edition

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

1. 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]

5. 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.