Thomas Calculus 13th [Solutions]

Section 8.3 Trigonometric Integrals 569
8.3 TRIGONOMETRIC INTEGRALS
1. cos 2x dx 1 cos 2x 2dx 1 sin 2x C
2 2
2. 3 sin x dx 9 sin x 1 dx 9 cos x 9 cos cos0 9 1 1 9
0 3 0 3 3 3 0
3 2 2
3.
4.
5.
3 3 1 4
cos x sin x dx cos x( sin x) dx cos x C
4
4 1 4 1 5
sin 2x cos 2x dx sin 2x cos 2x 2dx sin 2x C
2 10
3 2 2 2 1 3
sin x dx sin x sin x dx 1 cos x sin x dx sin x dx cos x sin x dx cos x cos x C
3
6.
7.
3 2 1 2 1 1 2
cos 4x dx cos 4x cos 4x dx 1 sin 4x cos 4x 4dx cos 4x 4dx sin 4x cos 4x 4dx
4 4 4
1 1 3
sin 4x sin 4x C
4 12
5 2
2
2
2
2 4
sin x dx sin x sin x dx 1 cos x sin x dx 1 2cos x cos x sin x dx
2 4 2 3 1 5
sin x dx 2cos x sin x dx cos x sin x dx cos x cos x cos x C
3 5
8.
5 2 4
sin x dx (using Exercise 7) sin x dx 2cos x sin x dx cos x sin x dx
0 2 0 2 0 2 2 0 2 2
3 5
2cos x 4 cos x 2 cos x (0) 2 4 2 16
2 3 2 5 2 0
3 5 15
9.
3 2 2 2 1 3
cos x dx cos x cos x dx 1 sin x cos x dx cos x dx sin x cos x dx sin x sin x C
3
10.
6 5 6 2
2 6 2
2
3cos 3x dx cos 3x cos3x 3dx 1 sin 3x cos3x 3dx
0 0 0
/6 2 4
1 2sin 3x sin 3x cos3x 3dx
0
6 6 2 6 4 sin 3 sin 3
0 cos 3 x 3 dx 2 0 sin 3 x cos 3 x 3 dx
0 sin 3 x cos 3 x 3 dx sin 3 x 2 x x
3 5
0
1 2 1 (0) 8
3 5 15
3 5 6
11.
3 3 3 2 3 2 3 5
sin x cos x dx sin x cos x cos xdx sin x 1 sin x cos x dx sin x cos x dx sin x cos x dx
1 4 1 6
sin x sin x C
4 6
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