Thomas Calculus 13th [Solutions]

568 Chapter 8 Techniques of Integration
2 2 2 2 2
1 x dx x 1 x 1 dx 1 x dx 2 1 x dx x 1 x 1 dx
2 2
1 x
1 x
2 2
1 x dx
x
1 x 1 1 dx C
2 2 2
1 x
71.
1 1 1 1 1
sin x dx x sin x sin y dy x sin x cos y C x sin x cos sin x C
72.
1 1 1 1 1
tan x dx x tan x tan y dy x tan x ln | cos y | C x tan x ln cos tan x C
73.
1 1 1
sec x dx x sec x sec y dy x sec x ln | sec y tan y|
C
1 1 1 1 2
x sec x ln sec sec x tan sec x C x sec x ln x x 1 C
74.
y
2 y
log2 x dx x log2 x 2 dy x log2 x C x log
x
ln 2 2 x C
ln 2
75. Yes,
cos
1
x is the angle whose cosine is x which implies
1 2
sin cos x 1 x .
76. Yes,
tan
1
x is the angle whose tangent is x which implies
1 2
sec tan x 1 x .
77. (a)
(b)
1 1 1 1 1
sinh x dx x sinh x sinh y dy x sinh x cosh y C x sinh x cosh sinh x C;
1 1 1 1 1
check: d x sinh x cosh sinh x C sinh x
x
sinh sinh x 1 dx sinh x dx
2 2
1 x
1 x
sinh 1 1 1 2
1 2
1 2
1 2
x dx x sinh x x 1 dx x sinh x 1
2
1
2
1 x 2 x dx x sinh x 1 x C
x
1 2
1 2
1 1
check: d x sinh x 1 x C sinh x
x x
dx sinh x dx
2 2
1 x 1 x
78. (a)
1 1 1
tanh x dx x tanh x tanh y dy x tanh x ln | cosh y | C
1 1
x tanh x ln cosh tanh x C;
check:
1 1 1
sinh tanh x
d x tanh x ln cosh tanh x C tanh x x
1 dx
2 1 2
1 x cosh tanh x 1 x
1
tanh
1 x x x dx tanh
1 x dx
2 2
1 x 1 x
(b) tanh 1 1 1 1 2
x dx x tanh x x dx tanh x 1 2x
dx x 1
2 2 tanh x
2
1 1
2
ln 1 x C
x
x
1 2 1 1
check: d x tanh x 1 ln 1 x C tanh x x x dx tanh x dx
2 2 2
1 x 1 x
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