168
CHAPTER 16
16.1
A table of integrals can be consulted to determine ax
a dx
cosh ln
1 tanh
=
∫
Therefore,
t d
d d
t d
d
t m
gc gc
m c
gm dt
t m
gc c
gm cosh
ln tanh
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎣ ⎡
⎟ ⎟
⎠ ⎞
⎜ ⎜
⎝ ⎛
= ⎟
⎟ ⎠
⎞ ⎜
⎜ ⎝
⎛
∫
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎣ ⎡
− ⎟
⎟ ⎠
⎞ ⎜
⎜ ⎝
⎛ cosh0
ln cosh
ln
2 2
t m
gc gc
gm
d d
Since cosh0 = 1 and ln1 = 0, this reduces to
⎟ ⎟
⎠ ⎞
⎜ ⎜
⎝ ⎛
t m
gc c
m
d d
cosh ln
16.2 a The analytical solution can be evaluated as
[ ]
1 3.50016773
5 .
5 .
4 5
. 1
2 4
2 4
2 4
2
= −
− +
= +
= −
− −
− −
∫
e e
e x
dx e
x x
b
single application of the trapezoidal rule 88
. 42
99329 .
1 2
999665 .
4 =
= +
−
t
ε
c
composite trapezoidal rule n = 2:
35 .
15 96303
. 2
4 999665
. 981684
. 2
4 =
= +
+ −
t
ε
n = 4:
47 .
4 3437
. 3
8 999665
. 99752
. 981684
. 86466
. 2
4 =
= +
+ +
+ −
t
ε
d
single application of Simpson’s 13 rule
169 17
. 6
28427 .
3 6
999665 .
981684 .
4 4
= =
+ +
−
t
ε
e
composite Simpson’s 13 rule n = 4 84
. 47059
. 3
12 999665
. 981684
. 2
99752 .
86466 .
4 4
= =
+ +
+ +
−
t
ε
f
Simpson’s 38 rule. 19
. 3
388365 .
3 8
999665 .
995172 .
930517 .
3 4
= =
+ +
+ −
t
ε
16.3 a The analytical solution can be evaluated as
[ ]
12.424778 sin
3 6
2 sin
3 2
6 sin
3 6
cos 3
6
2 2
= −
− +
= +
= +
∫
π π
π π
x x
dx x
b
single application of the trapezoidal rule 18
. 5
78097 .
11 2
6 9
2 =
= +
⎟ ⎠
⎞ ⎜
⎝ ⎛ −
t
ε π
c
composite trapezoidal rule n = 2:
25 .
1 26896
. 12
4 6
12132 .
8 2
9 2
= =
+ +
⎟ ⎠
⎞ ⎜
⎝ ⎛ −
t
ε π
n = 4:
3111 .
386125 .
12 8
6 14805
. 7
12132 .
8 77164
. 8
2 9
2 =
= +
+ +
+ ⎟
⎠ ⎞
⎜ ⎝
⎛ −
t
ε π
d
single application of Simpson’s 13 rule 0550
. 4316
. 12
6 6
12132 .
8 4
9 2
= =
+ +
⎟ ⎠
⎞ ⎜
⎝ ⎛ −
t
ε π
e
composite Simpson’s 13 rule n = 4 0032
. 42518
. 12
12 6
12132 .
8 2
14805 .
7 7716
. 8
4 9
2 =
= +
+ +
+ ⎟
⎠ ⎞
⎜ ⎝
⎛ −
t
ε π
f
Simpson’s 38 rule.
170 0243
. 42779
. 12
8 6
5 .
7 59808
. 8
3 9
2 =
= +
+ +
⎟ ⎠
⎞ ⎜
⎝ ⎛ −
t
ε π
16.4 a The analytical solution can be evaluated as
1104 3
2 2
2 2
2 3
4 4
2 4
4 3
2 2
4 1
6 4
2 6
4 2
4 2
6 4
2 4
2 5
3
= −
− −
+ −
+ −
− +
− −
= ⎥
⎦ ⎤
⎢ ⎣
⎡ +
− −
= +
− −
− −
∫
x x
x x
dx x
x x
b
single application of the trapezoidal rule
3 .
378 5280
2 1789
29 2
4 =
= +
− −
−
t
ε
c
composite trapezoidal rule n = 2:
6 .
138 2634
4 1789
2 2
29 2
4 =
= +
− +
− −
−
t
ε n = 4:
4 .
37 875
. 1516
8 1789
3125 .
131 2
9375 .
1 2
29 2
4 =
= +
+ −
+ +
− −
−
t
ε
d
single application of Simpson’s 13 rule
7 .
58 1752
6 1789
2 4
29 2
4 =
= +
− +
− −
−
t
ε
e
composite Simpson’s 13 rule n = 4
6685 .
3 5
. 1144
12 1789
2 2
3125 .
131 9375
. 1
4 29
2 4
= =
+ −
+ +
+ −
− −
t
ε
f
Simpson’s 38 rule. 09
. 26
1392 8
1789 31
1 3
29 2
4 =
= +
+ +
− −
−
t
ε
g
Boole’s rule. 1104
90 1789
7 3125
. 131
32 2
12 9375
. 1
32 29
7 2
4 =
= +
+ −
+ +
− −
−
t
ε
171
16.5 a
