74 Higher Engineering Mathematics
=
θ −
θ 3
18
+
θ 5
600
−
θ 7
7(5040)
+ · · ·
1
0
= 1 −
1
18
+
1
600
−
1
7(5040)
+ · · ·
= 0.946, correct to 3 significant figures.
Problem 15. Evaluate
0.4
0 x ln(1 + x) dx using
Maclaurin’s theorem, correct to 3 decimal places.
From Problem 6,
ln(1 + x) = x −
x 2
2
+
x 3
3
−
x 4
4
+
x 5
5
− · · ·
Hence
0.4
0
x ln(1 + x)dx
=
0.4
0
x
x −
x 2
2
+
x 3
3
−
x 4
4
+
x 5
5
− · · ·
dx
=
0.4
0
x
2
−
x 3
2
+
x 4
3
−
x 5
4
+
x 6
5
− · · ·
dx
=
x 3
3
−
x 4
8
+
x 5
15
−
x 6
24
+
x 7
35
− · · ·
0.4
0
=
(0.4) 3
3
−
(0.4) 4
8
+
(0.4) 5
15
−
(0.4) 6
24
+
(0.4) 7
35
− · · ·
− (0)
= 0.02133 − 0.0032 + 0.0006827 − · · ·
= 0.019, correct to 3 decimal places.
Now try the following exercise
Exercise 33 Further problems on
numerical integration using Maclaurin’s
series
1. Evaluate
0.6
0.2 3e sin θ dθ, correct to 3 decimal
places, using Maclaurin’s series.
[1.784]
2. Use Maclaurin’s theorem to expand cos2θ and
hence evaluate, correct to 2 decimal places,
1
0
cos 2θ
θ
1
3
dθ.
[0.88]
3. Determine the value of
1
0
√ θ cos θ dθ, correct to 2 significant figures, using Maclaurin’s
series.
[0.53]
4. Use Maclaurin’s theorem to expand
√
x ln(x + 1) as a power series. Hence
evaluate, correct to 3 decimal places,
0.5
0
√
x ln (x + 1) dx.
[0.061]
8.6 Limiting values
It is sometimes necessary to find limits of the form
lim
x→a
f (x)
g(x)
, where f (a) = 0 and g(a) = 0.
For example,
lim
x→1
x 2 + 3x − 4
x 2 − 7x + 6
=
1 + 3 − 4
1 − 7 + 6
=
0
0
and
0
0 is generally referred to as indeterminate.
For certain limits a knowledge of series can sometimes
help.
For example,
lim
x→0
tan x − x
x 3
≡ lim
x→0
⎧
⎪ ⎨
⎪ ⎩
x +
1
3
x 3 + · · · − x
x 3
⎫
⎪ ⎬
⎪ ⎭
from Problem 5
= lim
x→0
⎧
⎪ ⎨
⎪ ⎩
1
3
x 3 + · · ·
x 3
⎫
⎪ ⎬
⎪ ⎭
= lim
x→0
1
3
=
1
3
Similarly,
lim
x→0
sinh x
x
≡ lim
x→0
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
x +
x
3
3!
+
x
5
5!
+
x
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
from Problem 11
= lim
x→0
1 +
x 2
3!
+
x 4
5!
+ · · ·
= 1