Integration usingpartial fractions 413
=
1
2a
[ln(x − a) − ln(x + a)] + c
=
1
2a
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)
Let x = a then A =
1
2a
. Let x =−a then B =
1
2a
Hence
1
(a 2 − x 2 )
dx
=
1
2a
1
(a − x)
+
1
(a + x)
dx
=
1
2a
[−ln(a − x) + ln(a + x)] + c
=
1
2a
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 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
6x − 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 + 6θ
2
− 2θ
3
θ 2 (θ 2 + 1)
dθ
= 1.606, correct to 4 significant figures.