Higher Engineering Mathematics

412 INTEGRAL CALCULUS

=

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

−ln 1

]

= 1. 3412 , correct to 4

decimal places

Now try the following exercise.

Exercise 165 Further problems on integra-

tion 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.