Thomas Calculus 13th [Solutions]

574 Chapter 8 Techniques of Integration
46.
4 4 4 2 2 4 2 2 4 2
6 tan x dx 6 sec x 1 tan x dx 6 sec x tan x dx 6 tan x dx
4 4 4 4
4 2 2 4 2 tan
4 2
4
6 sec x tan x dx 6 sec x 1 dx 6 x 6 sec x dx 6 dx
4 4 3
4 4 4
3 4
4 4
2 1 ( 1) 6 tan x 6x
4 6 1 ( 1) 3 3 3 8
4 4 2 2
47.
5 4 2
2
4 2
tan x dx tan x tan x dx sec x 1 tan x dx sec x 2sec x 1 tan x dx
4 2
sec x tan x dx 2 sec x tan x dx tan x dx
3 1 4 2
sec x sec x tan x dx 2 sec x sec x tan x dx tan x dx sec x sec x ln |sec x|
C
4
2
2
1 2 1 4 1 2
tan x 1 tan x 1 ln |sec x| C tan x tan x ln|sec x|
C
4 4 2
48.
6 4 2 4 2 4 2 4
cot 2x dx cot 2x cot 2x dx cot 2x csc 2x 1 dx cot 2x csc 2x dx cot 2x dx
4 2 2 2 4 2 2 2
cot 2x csc 2x dx cot 2x cot 2x dx cot 2x csc 2x dx cot 2x csc 2x 1 dx
4 2 2 2 2
cot 2x csc 2x dx cot 2x csc 2x dx cot 2xdx
4 2 2 2 2
cot 2x csc 2x dx cot 2x csc 2xdx csc 2x 1 dx
4 2 2 2 2 1 5 1 3
cot 2x csc 2x dx cot 2x csc 2xdx csc 2x dx dx cot 2x cot 2x 1 cot 2x x C
10 6 2
49.
3 3 3 2 3 2 3
cot
6 cot x dx
6 csc x 1 cot x dx
6 csc x cot x dx
6 cot x dx x
2
ln |csc x |
6
2 3
1 1 3 ln 2 ln 2 4 ln 3
2 3 3
3
50.
4 2 2 2 2 2 8 3 2
8cot t dt 8 csc t 1 cot t dt 8 csc t cot t dt 8 cot t dt cot t 8 csc t 1 dt
3
8 3
cot 8 cot 8
3 t t t C
51. sin 3x cos 2 x dx 1 (sin x sin 5 x) dx 1 cos x 1 cos 5x C
2 2 10
52. sin 2x cos 3x dx 1 sin( x) sin 5x dx 1 sin x sin 5x dx 1 cos x 1 cos 5x C
2 2 2 10
53. sin 3x sin 3x dx 1 cos 0 cos 6x dx 1 dx 1 cos 6x dx 1 x 1 sin 6x
2 2 2 2 12
0
2 2
54.
2
1
2
1
2
1 2
sin x cos x dx sin 0 sin 2x dx sin 2x dx cos 2 x 1 ( 1 1) 1
0 2 0 2 0
4 0 4 2
55. cos 3x cos 4x dx 1 cos( x) cos 7x dx 1 cos x cos 7x dx 1 sin x 1 sin 7x C
2 2 2 14
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