Higher Engineering Mathematics, Sixth Edition

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]

5. 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