4. Trig Integrals
(1) Standard Integrals
1
sin axdx a cos ax c
1
cos axdx a sin ax c
1
sec 2 axdx tan ax c
a
5. Trig Integrals
(1) Standard Integrals
1
sin axdx a cos ax c
1
cos axdx a sin ax c
1
sec 2 axdx tan ax c
a
sin ax
tan axdx cos ax dx
6. Trig Integrals
(1) Standard Integrals
1
sin axdx a cos ax c
1
cos axdx a sin ax c
1
sec 2 axdx tan ax c
a
sin ax
tan axdx cos ax dx
1 1
log cos ax c OR log sec ax c
a a
9. 2 sin n x or cos n x
sin xdx cos x c
sin 2 xdx
10. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
11. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
12. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx
13. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx sin x sin xdx
2
14. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx sin x sin xdx
2
sin x1 cos 2 x dx
15. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx sin x sin xdx
2
sin x1 cos 2 x dx u cos x
du sin xdx
16. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx sin x sin xdx
2
sin x1 cos 2 x dx u cos x
du sin xdx
1 u 2 du
17. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx sin x sin xdx
2
sin x1 cos 2 x dx u cos x
du sin xdx
1 u 2 du
1 3
u u c
3
1 3
cos x cos x c
3
18. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx sin x sin xdx
2
sin x1 cos 2 x dx u cos x
du sin xdx
1 u 2 du
Odd Power
1 3
u u c Factorise as sin xsin x
2 some power
3
1 3 Substitute sin 2 x 1 cos 2 x
cos x cos x c
3 Use u cos x
21. sin xdx sin x dx
4 2 2
1
1 cos 2 x dx
2
4
22. sin xdx sin x dx
4 2 2
1
1 cos 2 x dx
2
4
1 2 cos 2 x cos 2 2 x dx
1
4
23. sin xdx sin x dx
4 2 2
1
1 cos 2 x dx
2
4
1 2 cos 2 x cos 2 2 x dx
1
4
1 2 cos 2 x 1 cos 4 x dx
1 1
4 2
24. sin xdx sin x dx
4 2 2
1
1 cos 2 x dx
2
4
1 2 cos 2 x cos 2 2 x dx
1
4
1 2 cos 2 x 1 cos 4 x dx
1 1
4 2
2 cos 2 x cos 4 x dx
1 3 1
4 2 2
25. sin xdx sin x dx
4 2 2
1
1 cos 2 x dx
2
4
1 2 cos 2 x cos 2 2 x dx
1
4
1 2 cos 2 x 1 cos 4 x dx
1 1
4 2
2 cos 2 x cos 4 x dx
1 3 1
4 2 2
x sin 2 x sin 4 x c
13 1
42 8
26. xdx sin x dx
Even Power
sin
4 2 2
Factorise as sin x
2 some power
1
1 cos 2 x dx
2
1
4 Substitute sin x 1 cos 2 x
2
2
1 2 cos 2 x cos 2 2 x dx
1
4
1 2 cos 2 x 1 cos 4 x dx
1 1
4 2
2 cos 2 x cos 4 x dx
1 3 1
4 2 2
x sin 2 x sin 4 x c
13 1
42 8
29. sin xdx sin xsin x dx
5 2 2
sin x1 cos x dx
2 2
30. sin xdx sin xsin x dx
5 2 2
sin x1 cos x dx u cos x
2 2
du sin xdx
31. sin xdx sin xsin x dx
5 2 2
sin x1 cos x dx u cos x
2 2
du sin xdx
1 u du
2 2
1 2u 2 u 4 du
32. sin xdx sin xsin x dx
5 2 2
sin x1 cos x dx u cos x
2 2
du sin xdx
1 u du
2 2
1 2u 2 u 4 du
u 2 u3 1 u5 c
3 5
2 1
cos x cos3 x cos 5 x c
3 5
34. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
35. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
e.g. i cos 5 x sin 3 xdx
36. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
e.g. i cos 5 x sin 3 xdx
cos5 x1 cos 2 x sin xdx
37. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
e.g. i cos 5 x sin 3 xdx
cos5 x1 cos 2 x sin xdx u cos x
du sin xdx
38. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
e.g. i cos 5 x sin 3 xdx
cos5 x1 cos 2 x sin xdx u cos x
u 5 1 u 2 du
du sin xdx
u 7 u 5 du
39. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
e.g. i cos 5 x sin 3 xdx
cos5 x1 cos 2 x sin xdx u cos x
u 5 1 u 2 du
du sin xdx
u 7 u 5 du
1 8 1 6
u u c
8 6
1 8 1
cos x cos 6 x c
8 6
40. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
e.g. i cos 5 x sin 3 xdx
cos5 x1 cos 2 x sin xdx u cos x
u 5 1 u 2 du
du sin xdx
u 7 u 5 du
Both powers odd
1 8 1 6
u u c Choose either as u
8 6
Usually the higher power
1 8 1
cos x cos 6 x c
8 6
42. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx
43. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
du cos xdx
44. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
45. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
1 7 1 9
u u c
7 9
1 7 1 9
sin x sin x c
7 9
46. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
47. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
iii sin 2 x cos 2 xdx
48. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
iii sin 2 x cos 2 xdx sin 2 x1 sin 2 x dx
49. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
iii sin 2 x cos 2 xdx sin 2 x1 sin 2 x dx
sin 2 x sin 4 x dx
50. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
iii sin 2 x cos 2 xdx sin 2 x1 sin 2 x dx
sin 2 x sin 4 x dx
1 1 3 1 1
x sin 2 x x sin 2 x sin 4 x c
2 4 8 4 32
51. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
iii sin 2 x cos 2 xdx sin 2 x1 sin 2 x dx
sin 2 x sin 4 x dx
1 1 3 1 1
x sin 2 x x sin 2 x sin 4 x c
2 4 8 4 32
1 1
x sin 4 x c
8 32
52. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
iii sin 2 x cos 2 xdx sin 2 x1 sin 2 x dx
sin 2 x sin 4 x dx
Both powers even
1 1 3 1 1
Use sin x 1 cos x
2 2 x sin 2 x x sin 2 x sin 4 x c
2 4 8 4 32
or cos 2 x 1 sin 2 x 1 1
x sin 4 x c
8 32
54. 4 tan n x or cot n x
tan xdx log cos x c
55. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx
2
56. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
57. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
tan x x c
58. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
tan x x c
tan 3 xdx
59. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
tan x x c
tan 3 xdx tan xsec 2 x 1dx
tan x sec 2 xdx tan xdx
60. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
tan x x c
tan 3 xdx tan xsec 2 x 1dx
tan x sec 2 xdx tan xdx u tan x
du sec 2 xdx
61. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
tan x x c
tan 3 xdx tan xsec 2 x 1dx
tan x sec 2 xdx tan xdx u tan x
udu tan xdx du sec 2 xdx
62. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
tan x x c
tan 3 xdx tan xsec 2 x 1dx
tan x sec 2 xdx tan xdx u tan x
udu tan xdx du sec 2 xdx
1
u 2 log cos x c
2
1 2
tan x log cos x c
2
64. tan 4 xdx tan 2 xsec 2 x 1dx
tan 2 x sec 2 xdx tan 2 xdx
65. tan 4 xdx tan 2 xsec 2 x 1dx
tan 2 x sec 2 xdx tan 2 xdx u tan x
du sec 2 xdx
66. tan 4 xdx tan 2 xsec 2 x 1dx
tan 2 x sec 2 xdx tan 2 xdx u tan x
u du tan xdx
2 2 du sec 2 xdx
67. tan 4 xdx tan 2 xsec 2 x 1dx
tan 2 x sec 2 xdx tan 2 xdx u tan x
u du tan xdx
2 2 du sec 2 xdx
1 3
u tan x x c
3
1 3
tan x tan x x c
3
70. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
71. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
72. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
73. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
74. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx
75. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx
76. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x
77. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x
du sec x tan xdx
78. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x
du sec x tan xdx dv sec 2 xdx
79. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x v tan x
du sec x tan xdx dv sec 2 xdx
80. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x v tan x
sec x tan x sec x tan 2 xdx du sec x tan xdx dv sec 2 xdx
81. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x v tan x
sec x tan x sec x tan 2 xdx du sec x tan xdx dv sec 2 xdx
sec x tan x sec xsec 2 x 1dx
sec x tan x sec3 xdx sec xdx
sec x tan x sec3 xdx logsec x tan x
82. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x v tan x
sec x tan x sec x tan 2 xdx du sec x tan xdx dv sec 2 xdx
sec x tan x sec xsec 2 x 1dx
sec x tan x sec3 xdx sec xdx
sec x tan x sec3 xdx logsec x tan x
2 sec3 xdx sec x tan x logsec x tan x c
83. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x v tan x
sec x tan x sec x tan 2 xdx du sec x tan xdx dv sec 2 xdx
sec x tan x sec xsec 2 x 1dx
sec x tan x sec3 xdx sec xdx
sec x tan x sec3 xdx logsec x tan x
2 sec3 xdx sec x tan x logsec x tan x c
1 1
sec3 xdx sec x tan x logsec x tan x c
2 2
84. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x v tan x
sec x tan x sec x tan 2 xdx du sec x tan xdx dv sec 2 xdx
sec x tan x sec xsec 2 x 1dx
Odd powers
sec x tan x sec xdx sec xdx
3
Done by parts
sec x tan x sec3 xdx logsec x tan x
2 sec3 xdx sec x tan x logsec x tan x c
1 1
sec3 xdx sec x tan x logsec x tan x c
2 2
87. sec 4 xdx sec 2 x1 tan 2 x dx u tan x
du sec 2 xdx
88. sec 4 xdx sec 2 x1 tan 2 x dx u tan x
1 u du
2 du sec 2 xdx
89. sec 4 xdx sec 2 x1 tan 2 x dx u tan x
1 u du
2 du sec 2 xdx
1 3
u u c
3
1 3
tan x tan x c
3
90. sec 4 xdx sec 2 x1 tan 2 x dx u tan x
1 u du
2 du sec 2 xdx
1 3 Even Power
u u c
Factorise as sec xsec x
3 2 2 some power
1 3
tan x tan x c Substitute sec 2 x 1 tan 2 x
3
Use u tan x
91. sec 4 xdx sec 2 x1 tan 2 x dx u tan x
1 u du
2 du sec 2 xdx
1 3 Even Power
u u c
Factorise as sec xsec x
3 2 2 some power
1 3
tan x tan x c Substitute sec 2 x 1 tan 2 x
3
Use u tan x
Exercise 2C; 1, 2, 4, 5, 7, 8, 10, 12, 14, 16, 17, 18