Thomas Calculus 13th [Solutions]

Section 7.3 Hyperbolic Functions 521
7.3 HYPERBOLIC FUNCTIONS
1.
sinh x 3 2
3
2
cosh x 1 sinh x 1 1 9 25 5 ,
4 4 16 16 4
coth x 1 5 , sech x 1 4 , and csch x 1 4
tanh x 3
cosh x 5
sinh x 3
3
4
5
4
tanh x sinh x
3
cosh x
5
,
2.
2
sinh x 4 cosh x 1 sinh x 1 16 25 5 ,
3 9 9 3
sech x 1 3 , and csch x 1 3
cosh x 5
sinh x 4
4
3
5
3
sinh x
tanh x 4 , coth x 1 5 ,
cosh x 5
tanh x 4
3.
cosh x 17 2
17
2
, x 0 sinh x cosh x 1 1 289 1 64 8 ,
15 15 225 225 15
coth x 1 17 , sech x 1 15 , and csch x 1 15
tanh x 8
cosh x 17
sinh x 8
8
15
17
15
tanh x sinh x 8
cosh x 17
,
4.
cosh x 13 2
, x 0 sinh x cosh x 1 169 1 144 12 ,
5 25 25 5
coth x 1 13 , sech x 1 5 , and csch x 1 5
tanh x 12
cosh x 13
sinh x 12
12
5
13
5
tanh x sinh x 12
cosh x 13
,
5.
ln x
ln x
lnx
2cosh (ln x) 2 e e e 1 x 1
2
ln x
e x
6.
sinh (2ln x)
2 1
2ln 2ln ln
2
ln
2 x
x x x x
x
2 4
2
e e e e x 1
2 2 2 2x
7.
5x 5x 5x 5x
e e e e 5x
cosh 5x sinh 5x e 8.
2 2
3x 3x 3x 3x 3x
cosh 3x sinh 3x e e e e e
2 2
9.
4
4
x x x x
x
4
4x
(sinh x cosh x)
e e e e e e
2 2
10.
2 2
ln(cosh x sinh x) ln(cosh x sinh x) ln cosh x sinh x ln1 0
11. (a) sinh 2x sinh( x x) sinh x cosh x cosh x sinh x 2sinh x cosh x
2 2
(b) cosh 2x cosh( x x) cosh x cosh x sinh x sin x cosh x sinh x
12.
2 2
2 2
x x x x
cosh sinh
1 x x x x x x x x
x x e e e e e e e e e e e e
2 2 4
1 1 0
2 x x
e 2e 4 e 1 (4) 1
4 4 4
dy
13. y 6sinh x 6 cosh x 1 2cosh x
3 dx
3 3 3
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