Thomas Calculus 13th [Solutions]

372 Chapter 5 Integration
34.
0 x 1 1
0
1 x 1 1 2
dx
( )
1 ln 1 ln
35.
36.
1 x 1 x 1 1 1 0
xe dx e e e 1 ( e 1)
0
2 2 2 2
0
2 2 1
2 2
2
ln x 1 1 2 1 2 1 2
1 x dx 2 (ln x)
1 2 (ln 2) 2 (ln1) 2
(ln 2)
37.
5 5
5
x
2 1/2 2
dx x(1 x ) dx 1 x 26 5
1 x
2
2 2 2
38.
/3 2 /3 2 3
/3
1 1 3 1 3 3
sin x cos x dx (sin x) cos x dx (sin x) sin sin (0)
0 0 3 0 3 3 3 8
x
x
39. (a) cos t dt [sin t]
0
0
(b)
d (sin x) cos x 1 x
dx
2
x
sin x sin 0 sin x d cos t dt
dx 0
1/2
d
x
1 1/2 cos x
cos t dt (cos x) d ( x) (cos x)
x
dx 0 dx 2
2 x
cos
2
x
x
40. (a)
(b)
sin x 2 3 sin 3 sin 2 3 2
3
1 t dt [ t x
] 1 sin x 1 d
x
3 d (sin 1) 3sin cos
dx 1
t dt dx
x x x
d
sin x 2 2 2
3 t dt (3sin x) d (sin x) 3sin x cos x
dx 1
dx
41. (a)
(b)
4 4 4
4
t 1/2 2 3/2 2 4 3/2 2 6 2 6 5
0 u du t
0 u du t
( ) 0 4
3 u d
t
d
0 3 t 3 t dt 0
u du dt 3
t t
d
dt
t
0
4
4 4 2 3 5
u du t d ( t ) t (4 t ) 4t
dt
42. (a)
(b)
tan 2 tan tan 2
0 sec y dy [tan y ] 0 tan (tan ) 0 tan (tan ) d
d 0
sec y dy
d
2 2
(tan(tan )) (sec (tan ))sec
d
d
tan 2 2 2 2
sec y dy (sec (tan )) d (tan ) (sec (tan ))sec
d 0
d
43. (a)
3 3
3
3
3 3
x
2
0 e t dt e t
x
e x x t x x
1 d
d 1 3
0
dx 0
e dt dx
e x e
(b)
d
dx
3 3 3
x 3 2
e t dt e x d ( x ) 3x e
x
0
dx
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