Thomas Calculus 13th [Solutions]

524 Chapter 7 Integrals and Transcendental Functions
1 2
40. If
1
dy
y x tanh x ln 1 x C , then
2
tanh 1 1 1 2
1
x x x
2 2
tanh x
dx , which verifies the
2
1 x 1 x
formula
41. sinh 2x dx 1 sinh u du , where u 2x and du 2 dx
2
cosh u cosh 2x
C C
2 2
42.
43.
sinh x dx 5 sinh u du, where u x and du 1 dx
5
5
5
5cosh u C 5cosh x C
5
6cosh x ln 3 dx 12 cosh u du , where
2
u x ln 3 and du 1 dx
2
2
12 sinh u C 12sinh x ln 3 C
2
44. 4cosh (3x ln 2) dx 4 cosh u du , where u 3x ln 2 and du 3 dx
3
4 sinh u C 4 sinh(3x ln 2) C
3 3
45.
sinh
tanh x
u
dx 7 du , where u x and du 1 dx
7 cosh u
7
7
x/7 x/7 x/7 x/7
7 ln | cosh u | C1 7 ln cosh x C
7 1 7 ln e e C
2 1 7ln e e 7 ln 2 C1
/7 /7
7 ln e x e x C
46. coth d 3 cosh u du
3
sinh u
, where
u and
3
du
d
3
/ 3 / 3
3 ln sinh u C1 3 ln sinh C
3
1 3 ln e e C
2 1
/ 3 / 3 / 3 / 3
3 ln e e 3 ln 2 C1
3 ln e e C
47.
48.
2 1
2
sech x dx sech u du , where u x 1 and du dx
2
2
tanh u C tanh x 1 C
2
2 2
csch (5 x) dx csch u du , where u (5 x ) and du dx
( coth u) C coth u C coth (5 x)
C
49.
sech
t tanh
t
t
dt 2 sech u tanh u du , where
2( sech u) C 2 sech t C
1/2
u t t and
du
dt
2 t
50.
csch (ln t) coth (ln t)
dt csch u coth u du , where u ln t and du dt
t
t
csch u C csch(ln t)
C
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