Thomas Calculus 13th [Solutions]

572 Chapter 8 Techniques of Integration
Let u 1 sin x u 1 sin x du cos x dx, x 5 u 1 sin 5 3 , x
6 6 2
u 1 sin 1
2 3 2 1 2 2
3 2 1 5 2 3 2
3 x ) 5 6 3 2 u 1) u du
3
x ) u
5 6 3 2
u u du
3 2
3 2
7 2 5 2 3 2
1
2 (1 sin ) 2 1 sin 5
2 u 4 u 2 u
3 3 6 7 5 3 3 2
2 2 3
3 2
2 4 2 2 3
7 2
4 3
5 2
2 3
3 2
4 3
5 2
2 3
7 2
18
3 3 2 7 5 3 7 2 5 2 3 2 5 2 7 2 35
30.
7 12 7 12 1 sin 2 7 12 2 7 12 2
1 sin 2 x 1 sin 2x cos 2x
1 sin 2x dx x dx dx dx
2 2 1 1 sin 2x 2 1 sin 2x 2 1 sin 2x
7 12 cos 2x
7 12
7 1 1 2 1
2 dx 1 sin 2 x 1 sin 2 1 sin 2 1 1
1 sin 2x 2
12 2 2
2 2
31.
2 2 2 2
0 1 cos 2 d
0 2 | sin | d 2 0
sin d 2 cos sin 0
2(1) 2
32.
33.
34.
2
3 2
2
3 2
3 0 3 3
1 cos t dt sin t dt sin t dt sin t dt sin t dt
0
0 2 2
1 cos t sin t dt 1 cos t sin t dt
0
0 0 2 2
sin t dt cos t sin t dt sin t dt cos t sin t dt
0 0
3 0
3
cos t cos t cos t cos t 1 1 1 1 1 1 1 1 8
3 3
0
3 3 3 3 3
2 2 1 2
sec x tan x dx tan x sec x dx tan x C
2
2 2
sec x tan x dx sec x tan x tan x dx; u tan x, du sec x dx, dv sec x tan x dx, v sec x;
3 2 2
sec x tan x sec x dx sec x tan x sec x sec x dx sec x tan x tan x 1 sec x dx
2 2
sec x tan x tan x sec x dx sec x dx sec x tan x ln | sec x tan x | tan x sec x dx
2 2
sec x tan x dx sec x tan x ln | sec x tan x | tan x sec x dx
2
2 tan x sec x dx sec x tan x ln | sec x tan x |
2
tan x sec x dx 1 sec x tan x 1 ln | sec x tan x | C
2 2
35.
36.
3 2 1 3
sec x tan x dx sec x sec x tan x dx sec x C
3
3 3 2 2 2 2
sec x tan x dx sec x tan x sec x tan x dx sec x sec x 1 sec x tan x dx
4 2 1 5 1 3
sec x sec x tan x dx sec x sec x tan x dx sec x sec x C
5 3
37.
2 2 2 2 1 3
sec x tan x dx tan x sec x dx tan x C
3
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