Thomas Calculus 13th [Solutions]

658 Chapter 8 Techniques of Integration
40. ln ax ( )
2
x
1
x
( )
1
3
3 x 3
1 3 1 1 3 1 3
ln
x
ln
3 x 3 3 9
I x ax dx x ax x C
41.
2dz
1
2
dx
z
dz
1 sin x
2
2z
1
1
z
1 z
2
2 2
C
2
(1 ) z 1 tan
x
2
C
42.
dx
2dz
1 z
2
2 dz dz x
x x 2
2 2
2z
1 z
1
1 z 2z 1 z z
1 z
2
1 z
2
ln |1 z | C ln tan 1 C
1 sin cos 1 2
43.
44.
2dz
1
2
dx
z
dz
1 sin x
2
2z
(1 ) 1
1
z
1 z
2
/2 1 1 2 2
1
0 0 0 z 0
2dz
1 z
2
1 z
2
1 z
2
/2
dx
1 1
dz 1
1
/3 1 cos x 1/ 3 1/ 3
2
1
z z 1/ 3
(1 2) 1
3 1
45.
46.
2 dz
1
2
1
d
z
2 dz 2 dz 2 1 1 3
tan
z 2
tan
1
2 cos 2
2 2 2
1 2 2 1 3 9
2
z
z z z
0
1 z
2
/2 1 1 1
0 0 0 0 3 3 3 3 3 3
1 z
2 2 dz
2
d 1
2
1
2
2 1 z dz
z z
2 2
z
dz ln z
z
3 3
2z
1 z
2
z z z z
z
2z
2
2
1
2
1 z
z
2 /3 3 3 3
3
cos 1 1
/2 sin cos sin 1 1 2 2 2 2 1 2 2 4
1
47.
1 ln 3 3 0 1 ln 3 1 1 (ln 3 2) 1 ln 3 1
2 4 4 4 2 4 2
2dz
1
2
dt z
dz dz
z
ln
t t 2
2 2
2z
1 z z z z
z
1 z
2
1 z
2
1 2
tan 1 2
2 2 1 C 1 ln
sin cos 2 1 ( 1) 2 2 1 2 2 tan 1 2
t
2
t
2
C
48.
49.
50.
1 z
2
2dz
2 2 2
t dt 1
2
1
2
2 1 z dz 2 1 z dz 1 z dz
z z
dz dz
t
2 2
2
2 2
2 2 2 2 2 2 2
2
1 z
1 1 z 1 z 1 z 1 z 1 z 1 z 1 z z z 1 z z
1 z
2
cos
1 cos 1
1
1
z C t C
dz 2 dz 2 tan cot t
2 2
z z z
2
1
2 dz
d 1
2
1 tan
d z 2 dz 2dz dz dz
2
cos 2
2
1 z 1 (1 z)(1 z) 1 z 1 z
z z C C
z
1 tan
2
2
1 z
sec ln |1 | ln |1 | ln
2 dz
1 z
2
csc d
d
dz
ln | z | C ln tan C
sin z
2
2z
1 z
2
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