Thomas Calculus 13th [Solutions]

528 Chapter 7 Integrals and Transcendental Functions
79.
2 2 2
2
0 0
V cosh x sinh x dx 1 dx 2
80.
ln 3 2
ln 3 3 1/ 3
V 2 0 sech x dx 2 tanh x
0
2 3 1/ 3
81.
1
ln 5 2 ln 5
1
ln 5
y cosh 2x y sinh 2x L 1 (sinh 2 x) dx cosh 2x dx sinh 2x
2 0 0 2 0
ln 5
2x
2x
1 e e
1 5 1 6
2 2 4 5 5
0
82. (a)
(b)
(c)
(d)
(e)
x 1 x 1 1 1
x x e
e
x
1
e
x e e
x
e
2x
x
e e
x x x 1
x
1 1
x x e e x e
1
1
x x e e
x x
e
2x
e
e
x
lim tanh lim lim lim lim 1 0
1 0
1
x 1
x 1
x x e
e
x 2x
e
x e
x
e e x e e
x x x 1
x
x 2x
x x e e x e 1 e e
e
x x e
x
e
x
lim tanh lim lim lim lim 1 0 1
1 0 1
1
x
x x
1
x
e e
e
ex e
lim sinh x lim lim lim 1 0
x x
2
x
2
x
2 2ex
x x x x
lim sinh x lim
e e
lim
e e
0
x x
2
x
2 2
1 2
2 2 e
x
e
x
x
x x x 1 1
1
1
x x e e x e x
2x
e x x
e e
lim sech lim lim lim 0
1 0
0
(f )
(g)
(h)
x 1 x 1 1 1
x x e
e
x
1
e
x e e
x
e
2x
x
e e
x x x 1
x
1 1
x x e e x e
1
1
x x e e
x x
e
2x
e
e
x
x 1
lim coth lim
x x lim e
x lim
2x
e
x
x e e e e 1
x x x 1 x 2x
x 0 x 0 e e x 0 e e 0 e 1
e
x x
x 1
lim coth lim
x x lim e
x lim
2x
e
x
x e e e e 1
x x x 1 x 2x
x 0 x 0 e e x 0 e e 0 e 1
e
x x
lim csch lim 2 lim 2 . x
lim x
x
e
2e
0
1
1 0 1
0
x x x x x
x x e e x e e e
e
x
x
lim coth lim lim lim lim 1 0
1 0
1
(i)
2
dy
83. (a) y H cosh w x tan H w sinh w x sinh w x
w H dx w H H H
(b) The tension at P is given by T
H cosh w x w H cosh w x wy
H w H
2
cos H T H sec H 1 tan H 1 sinh w x
H
2
84.
s 1 1 1 1
sinh sinh sinh sinh ;
a
ax ax as ax as x a
as
1 2 2 2 2
sinh ax 1 1 a s 1 s 1
a a 2
a
y 1 1 2
cosh cosh
a
ax a
ax
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