Higher Engineering Mathematics

INTEGRATION USING PARTIAL FRACTIONS 409

H

By dividing out and resolving into partial fractions
it was shown in Problem 4, page 20:

x^3 − 2 x^2 − 4 x− 4

x^2 +x− 2

≡x− 3 +

4

(x+2)

−

3

(x−1)

Hence

∫ 3

2

x^3 − 2 x^2 − 4 x− 4

x^2 +x− 2

dx

≡

∫ 3

2

{

x− 3 +

4

(x+2)

−

3

(x−1)

}

dx

=

[

x^2

2

− 3 x+4ln(x+2)−3ln(x−1)

] 3

2

=

(

9

2

− 9 +4ln5−3ln2

)

−(2− 6 +4ln4−3ln1)

=− 1. 687 , correct to 4 significant figures

Now try the following exercise.

Exercise 163 Further problems on integra-

tion using partial fractions with linear factors

In Problems 1 to 5, integrate with respect tox

1.

∫

12

(x^2 −9)

dx

⎡

⎢

⎣

2ln(x−3)−2ln(x+3)+c

or ln

{

x− 3

x+ 3

} 2

+c

⎤

⎥

⎦

2.

∫

4(x−4)

(x^2 − 2 x−3)

dx

⎡

⎢

⎣

5ln(x+1)−ln (x−3)+c

or ln

{

(x+1)^5

(x−3)

}

+c

⎤

⎥

⎦

3.

∫

3(2x^2 − 8 x−1)

(x+4)(x+1)(2x−1)

dx

⎡

⎢

⎢

⎢

⎣

7ln(x+4)−3ln(x+1)

−ln (2x−1)+c or

ln

{

(x+4)^7

(x+1)^3 (2x−1)

}

+c

⎤

⎥

⎥

⎥

⎦

4.

∫

x^2 + 9 x+ 8

x^2 +x− 6

dx

[

x+2ln(x+3)+6ln(x−2)+c

orx+ln{(x+3)^2 (x−2)^6 }+c

]

5.

∫

3 x^3 − 2 x^2 − 16 x+ 20

(x−2)(x+2)

dx

⎡

⎣

3 x^2

2

− 2 x+ln (x−2)

−5ln(x+2)+c

⎤

⎦

In Problems 6 and 7, evaluate the definite inte-

grals correct to 4 significant figures.

6.

∫ 4

3

x^2 − 3 x+ 6

x(x−2)(x−1)

dx [0.6275]

7.

∫ 6

4

x^2 −x− 14

x^2 − 2 x− 3

dx [0.8122]

8. Determine the value of k, given that:
   ∫ 1

0

(x−k)

(3x+1)(x+1)

dx= 0

[

1

3

]

9. The velocity constantkof a given chemical
   reaction is given by:

kt=

∫ (

1

(3− 0. 4 x)(2− 0. 6 x)

)

dx

wherex=0 when t = 0. Show that:

kt=ln

{

2(3− 0. 4 x)

3(2− 0. 6 x)

}

41.3 Worked problems on integration

using partial fractions with

repeated linear factors

Problem 5. Determine

∫

2 x+ 3

(x−2)^2

dx.

It was shown in Problem 5, page 21:

2 x+ 3

(x−2)^2

≡

2

(x−2)

+

7

(x−2)^2