Thomas Calculus 13th [Solutions]

Section 8.3 Trigonometric Integrals 575
56.
2
1
2
1 1 1
2
2 cos 7 x cos x dx
2 2 cos 6 x cos 8 x dx
2 6 sin 6 x
8
sin 8 x 0 2
57.
58.
2 1 cos 2
sin cos 3 d cos 3 d 1 cos 3 d 1
2 2 2
cos 2 cos 3 d
1 cos 3 d 1 1 cos(2 3) cos(2 3) d 1 cos 3 d 1
2 2 2 2 4
cos( ) cos 5 d
1 cos 3 d 1 cos d 1 cos 5 d 1 sin 3 1 sin 1 sin 5
2 4 4 6 4 20
C
2
cos 2 sin d
2
2cos
2
1 sin d
4
4cos
2
4cos 1 sin d
4 2 4 5 4 3
4cos sin d 4cos sin d sin d cos cos cos C
5 3
59.
60.
3 3 4 2 5
cos sin 2 d cos 2sin cos d 2 cos sin d cos C
5
3 2 2 2
sin cos 2 d sin cos 2 sin d 1 cos 2cos 1 sin d
4 2 4 2
2cos 3cos 1 sin d 2 cos sin d 3 cos sin d sin d
2 5 3
cos cos cos C
5
61. sin cos cos 3 d 1 2sin cos cos 3 d 1 sin 2 cos 3 d
2 2
1 1 sin(2 3) sin(2 3) d 1 sin( ) sin 5 d 1 sin sin 5 d
2 2 4 4
1 cos 1 cos 5 C
4 20
62. sin sin 2 sin 3 d 1 cos(1 2) cos(1 2) sin 3 d 1 cos( ) cos 3 sin 3 d
2 2
1 sin 3 cos d 1 sin 3 cos 3 d 1 1 sin(3 1) sin(3 1) d 1 2sin 3 cos 3 d
2 2 2 2 4
1 sin 2 sin 4 d 1 sin 6 d 1 cos 2 1 cos 4 1 cos 6 C
4 4 8 16 24
63.
2
2
2
3 sec sec tan x 1 sec x
sec x
x x tan x sec x sec x
dx dx dx dx dx tan x sec x dx csc x dx
tan x tan x tan x tan x tan x
sec x ln |csc x cot x|
C
64.
2
2
2
3 sin sin 1 cos x sin x
sin x
x x sin x cos x sin x
3
dx dx dx dx dx sec x tan x dx sec x tan x dx
4 4 4 4 4
cos x cos x cos x cos x cos x
2 1 3
sec x sec x tan x dx sec x tan x dx sec x sec x C
3
65.
2
2 2 1 cos x
2
1
2 2 2 2
tan x dx sin x sin x dx sin x dx sin x dx cos x sin x dx
csc x cos x cos x cos x cos x
sec x tan x dx sin x dx sec x cos x C
cot x cos x
66. dx 1 dx 2 dx 2 dx csc 2x 2dx ln |csc 2x cot 2 x|
C
2 sin 2
cos x
x cos x 2sin x cos x sin 2x
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