Thomas Calculus 13th [Solutions]

370 Chapter 5 Integration
10. (1 cos
0 x ) dx [ x sin x ] 0 ( sin ) (0 sin 0)
11.
3 /4 3 /4
csc cot 3
/4 d [ csc ] /4 csc csc 2 2 0
4 4
12.
/3
/3
sin u 4 4 4
4
2 du
4
0 cos u cos u 0 (1/2) 1
13.
0
1 cos 2 t 0
1 1 1 1
0
dt
2 2 /2 2 2 cos 2 t dt t
2 4
sin 2 t 1 (0) 1 sin 2(0) 1 1 sin 2
2 2 4 2 2 4 2 4
14.
/3 2
/3 sin t dt Use the double angle formula 2
cos2t
1 2sin t which implies that
2 1 cos(2 t)
sin t .
2
/3 2
sin t dt
/3
/3
/3
1 cos2t t sin 2t
dt
/3 2 2 4 /3
1 3 1 3 3
6 4 2 6 4 2 3 4
15.
/4 2 /4 2 /4
tan x dx (sec x 1) dx [tan x x] 0 0
0 tan (tan(0) 0) 1
4 4 4
16.
17.
/6 2 /6 2 2
/6 2
(sec x tan x) dx (sec x 2sec x tan x tan x)
dx (2sec x 2sec x tan x 1) dx
0 0
0
/6
[2 tan x 2sec x x] 0 2 tan 2sec (2 tan 0 2sec0 0) 2 3 2
6 6 6
6
/8
1
/8
1 1
2 2
sin 2x dx cos 2x
cos 2 cos 2(0)
0 2 0 2 2 4
4 2 4 2 2
4
18. 4sec t dt (4sec t t ) dt 4 tan t
3
2
t
/3 t 3
4 tan 4 tan (4( 1) 4) 4 3 3 4 3 3
4 3
4 3
19.
1 3 1
3
2 1 2 2 ( 1) 2
3
2
( r 1) dr ( r 2r 1) dr r r r
( 1) ( 1) 1 1 1 8
1 1 3
1
3 3 3
20.
3 2 3 3 2 2
3 ( t 1)( t 4) dt
3 ( t t 4 t 4) dt t t
4 3
2 t 4 t
3
4 3
3 3
4 3
2
2( 3) 4 3
4
3
3
4
3
3
4 3 3
2
2( 3) 4( 3) 10 3
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