Xii Indefinite Integrals Assignment
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ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e x log a ea log x e a log a
5.
sin 6 x cos6 x sin 2 x cos2 x
2.
1 1 cos x
6. tan 1
3.
1 sin x 1 sin x
7.
16.
sin 2 x a sin x b 2 cos 2 x
20.
1 e 1
1 cos x 1 cos x
18.
1 sin x a sin x b
21.
x
22.
x 1
25. sin 5 x
9 25x
2
ex 33. 2 x e 6e x 5 1
37.
a b cos x
16 6x x
2
23.
19.
1 cot x 1 cot x
sec 2 2 tan 1 x 1 x
2
2
x
24. sin 2 x cos5 x
30.
x4 1 x2 1
31.
1 3 2x x 2
32.
34.
1 x x n 1
35.
x x x2 1
36.
39.
2x 3 x 3 x 18
40.
38.
2
cos5 x 2 x 1 27. x log x 28. sin x x
26. tan 3 x
1
29.
15.
2
x3 x2
e x e x e x e x
14.
sin 2 x
8.
x4 3 x2 1
13. sin 4 x cos 7 x
1
1 3x 4 3x 1
12.
10. cos4 x
sin x sin x a
4. tan 1 (sec x tan x)
11. cos 2 x cos 4 x
9. sin 3 x cos3 x
17.
sin x 1 sin x
2x 2
1 x x
4
4
2
1 x 8 x 20 2
1 9 8x x 2
2sin 2 cos 6 cos 2 4sin
41.
x2 2 2 x 6x 5
42.
x2 x 2 6 x 12
43.
45.
1 a sin x b 2 cos 2 x
46.
sin x sin 3x
47.
1 2 3cos 2x
49.
1 1 2sin x
50.
1 5 4cos x
51.
1 3sin x 2cos x 52. 3 2sin x cos x 3cos x 2sin x
55.
x sin x 1 cos x
2
2
53. log x
2
54. sin 1 x
ax a x
44.
48.
3x 1 5 2 x x2
sin 2 x sin x cos4 x 4
56. sec3 x
57.
sin 1 x 2 32
1 x
1 sin x 61. e x 1 cos x
ax
62.
65. e cos bx
66.
69. x 1 1 x x 2
70.
73.
8
x 2 x2 4
85.
tan x
89.
82.
1
x 3
x 1
x 3 2 x x x 1
93.
97. e
x2 a 2 x
x
cos e x
74.
log x
1 log x
63.
2
7 x 10 x
2
86.
90.
94.
67.
2x
x2
x
2
1 x 2 4
cot x
75.
2
1 x
ex
16 log x
71.
sin 1 x x2
64.
1 1 log x log x 2
68. 3 x 2 x 2 x 1
1 sin x sin 2 x
tan tan 3 1 tan 3
x2 4 79. 4 x 16
83.
60.
2
x
2 x 1 x 1 x 2 x 3
x2 1 78. 4 2 x x 1
1 77. x x 5 1
81.
2x 59. tan 1 2 1 x
58. x 2 tan 1 x
1 4 sin x cos4 x
3x 1 2
x 2 x 2
76.
sin 2 x 1 sin x 2 sin x
80.
1 x 1
84.
x 2 1 x4 1
4
1 x 4 x 1
87.
sin x sin x
91.
x 1 e x 2 x 2
92.
95.
1 x 5 x 2 16
96. 1 2 tan x sec x tan x
99.
1 cos 2 x 2sin 2 cos 100. tan 1 2 6 cos 4sin 1 cos 2 x
2
ex 5 4e x e 2 x
x
98. cos log x
1
72.
x 1
4
2
1
88.
x 1
x
2
1 x 2
1 tan x x log cos x
ANSWERS (INDEFINITE INTEGRALS) ( add a constant c to every answer) ax x a 1 1. aa x log a a 1 6.
x x2 4 2
10.
7.
2. cot x cos ecx
2 32 32 3x 4 3x 1 27
1 sin 4 x 3 x 2sin 2 x 8 4
x 14. 2cot x 2
11.
8.
3. sec x tan x
x3 x 2 4 x 8log x 2 3
1 sin 6 x sin 2 x 2 6 2
15. log e x e x
16.
1 log a 2 sin 2 x b 2 cos 2 x 2 a b 2
20. log 1 e x
22.
sin 3 x 2sin 5 x sin 7 x 24. 3 5 7
27.
1 3 x log x 3
31.
1 x 1 log 4 3 x
28.
32.
x4 36. sin 1 5
3co s5 x 1 3 25. cos x cos x cos7 x 5 7
1 x4 tan 1 2 2
33.
x3 42. x 3log x 2 6 x 12 2 3 tan 1 3
49.
52.
46.
1 tan( x / 2) 2 3 log 3 tan( x / 2) 2 3 5 x 12 log 3cos x 2sin x 13 13
1 2 3
log
50.
19. log cos x sin x
41.
26.
1 1 5 x sin 5 3
34.
1 xn log n n x 1
2x2 1 38. sin 1 5
x3 37. sin 1 5
1 a tan x tan 1 ab b
29.
1 ex 1 log x 4 e 5
40. 2log sin 2 4sin 5 7 tan 1 sin 2
45.
sin x a sin x b
2 a log a b cos x 2 b a b cos x
1 4 sin x sin 2 x log sin x 4
1 3 1 cos 2 x cos 6 x 32 2 6
x3 1 1 x 4 tan 1 x 13. cos11x cos 3 x 3 22 6
18. cos ec a b .log
x 1
9.
5. tan x cot x 3 x
12.
17. sin a log sin x a x a cos a
21. 2log
x x2 4. 4 4
1 tan 2 tan 1 x 2
23.
1 tan 2 x log sec x 2
30.
x3 x 2 tan 1 x 3
35.
2 x2 1 1 tan 1 3 3
2 x3 39. log x 2 3 x 18 log 3 x6
1 1 log 2 x 2 6 x 5 tan 1 2 x 3 4 2
x 43. a sin 1 a 2 x 2 a
x 1 44. 3 5 2 x x 2 2sin 1 6
3 tan x 3 tan x
5 tan x 1 5 tan x 1
47.
1 2 5
2 tan x / 2 tan 1 3 3 2
53. x log x 2 x log x x
log
48. tan 1 tan 2 x
x 51. tan 1 1 tan 2 54. x sin 1 x 1 x 2
55. x cot
58.
x 2
1 1 sec x tan x log sec x tan x 2 2
x3 1 1 tan 1 x x 2 log x 2 1 3 6 6
61. e x cot
66.
56.
x 2
62.
59. 2 x tan 1 x log 1 x 2
x log x 1
63.
1 9 2x 7 2 x 7 7 x 10 x2 sin 1 4 8 3
ex 1 x
64.
67.
1 log x 2
x
57.
1 sin 1 x | log 1 x 2 2 1 x2
60.
1 1 x2 sin 1 x log x x
x log x
log x
65.
2
e ax a cos bx b sin bx a2 b2
16 8log log x
log x
2
16
68. x 2 x 13 2 7 x 1 x 2 x 1 3 log x 1 x 2 x 1 69. 1 1 x x 2 3 2 1 2 x 1 1 x x 2 5 sin 1 2 x 1 3 8 16 2 2 8 2 5
1 1 1 1 1 2 70. log x 1 1 cos x log x 2 log x 3 71. log 1 cos x log 1 cos x log 1 2cos x 6 3 2 2 6 3
72.
5 7 5 log x 2 log x 2 16 4 x 2 16
1 2 1 x x 73. log x 2 log x 2 4 tan 1 74. tan 1 x tan 1 2 2 3 3 2
1 1 1 2 tan 1 75. log 1 tan log tan 2 tan 1 tan 1 3 6 3 3
4
2 sin x 76. log 2 1 sin x
77.
x2 4 2 1 1 1 x2 2 x 1 1 x5 1 x2 x 1 1 x 1 78. log 2 79. tan 1 80. tan log log 5 5 x 1 2 x x 1 2 2 2 2 x 2 2 x 1 2 2 x 2x 4 2
81.
cot 1 tan 1 cot 2 cot 1 tan 2 tan 1 1 1 1 1 82. tan 1 log tan 1 log 2 cot 2 2 2 tan 2 2 2 cot 2 cot 1 2 tan 2 tan 1
2 2 83. 1 tan 1 tan x 1 84. 1 log x 2 x 1 85. 2 x tan 1 x 86. 1 log x 1 3 1 tan 1 x 1 2 2 x 2 2 x 1 2 4 3 x 1 3 2 2 tan x
87.
1 x2 x 1 1 1 1 88. 89. log x 2 1 tan 1 x log x 1 90. x cos 2 sin 2 .log sin x x 1 x 4 2 2
91.
ex x2
93.
a x2 x x 2 a 2 a log x
95.
x2 4 1 x 2 13 x 4 tan 1 log 8 3 x 2 13 x 4 3 x 16 3
96. log sec 2 x sec x tan x
98.
x sin log x cos log x 2
92. log x log cos x
1
ex 2 94. sin 1 3
97. 2sin e
99. 2log sin 2 4sin 5 7 tan 1 sin 2
100.
x
x2 2
5.
sin 6 x cos6 x sin 2 x cos2 x
2.
1 1 cos x
6. tan 1
3.
1 sin x 1 sin x
7.
16.
sin 2 x a sin x b 2 cos 2 x
20.
1 e 1
1 cos x 1 cos x
18.
1 sin x a sin x b
21.
x
22.
x 1
25. sin 5 x
9 25x
2
ex 33. 2 x e 6e x 5 1
37.
a b cos x
16 6x x
2
23.
19.
1 cot x 1 cot x
sec 2 2 tan 1 x 1 x
2
2
x
24. sin 2 x cos5 x
30.
x4 1 x2 1
31.
1 3 2x x 2
32.
34.
1 x x n 1
35.
x x x2 1
36.
39.
2x 3 x 3 x 18
40.
38.
2
cos5 x 2 x 1 27. x log x 28. sin x x
26. tan 3 x
1
29.
15.
2
x3 x2
e x e x e x e x
14.
sin 2 x
8.
x4 3 x2 1
13. sin 4 x cos 7 x
1
1 3x 4 3x 1
12.
10. cos4 x
sin x sin x a
4. tan 1 (sec x tan x)
11. cos 2 x cos 4 x
9. sin 3 x cos3 x
17.
sin x 1 sin x
2x 2
1 x x
4
4
2
1 x 8 x 20 2
1 9 8x x 2
2sin 2 cos 6 cos 2 4sin
41.
x2 2 2 x 6x 5
42.
x2 x 2 6 x 12
43.
45.
1 a sin x b 2 cos 2 x
46.
sin x sin 3x
47.
1 2 3cos 2x
49.
1 1 2sin x
50.
1 5 4cos x
51.
1 3sin x 2cos x 52. 3 2sin x cos x 3cos x 2sin x
55.
x sin x 1 cos x
2
2
53. log x
2
54. sin 1 x
ax a x
44.
48.
3x 1 5 2 x x2
sin 2 x sin x cos4 x 4
56. sec3 x
57.
sin 1 x 2 32
1 x
1 sin x 61. e x 1 cos x
ax
62.
65. e cos bx
66.
69. x 1 1 x x 2
70.
73.
8
x 2 x2 4
85.
tan x
89.
82.
1
x 3
x 1
x 3 2 x x x 1
93.
97. e
x2 a 2 x
x
cos e x
74.
log x
1 log x
63.
2
7 x 10 x
2
86.
90.
94.
67.
2x
x2
x
2
1 x 2 4
cot x
75.
2
1 x
ex
16 log x
71.
sin 1 x x2
64.
1 1 log x log x 2
68. 3 x 2 x 2 x 1
1 sin x sin 2 x
tan tan 3 1 tan 3
x2 4 79. 4 x 16
83.
60.
2
x
2 x 1 x 1 x 2 x 3
x2 1 78. 4 2 x x 1
1 77. x x 5 1
81.
2x 59. tan 1 2 1 x
58. x 2 tan 1 x
1 4 sin x cos4 x
3x 1 2
x 2 x 2
76.
sin 2 x 1 sin x 2 sin x
80.
1 x 1
84.
x 2 1 x4 1
4
1 x 4 x 1
87.
sin x sin x
91.
x 1 e x 2 x 2
92.
95.
1 x 5 x 2 16
96. 1 2 tan x sec x tan x
99.
1 cos 2 x 2sin 2 cos 100. tan 1 2 6 cos 4sin 1 cos 2 x
2
ex 5 4e x e 2 x
x
98. cos log x
1
72.
x 1
4
2
1
88.
x 1
x
2
1 x 2
1 tan x x log cos x
ANSWERS (INDEFINITE INTEGRALS) ( add a constant c to every answer) ax x a 1 1. aa x log a a 1 6.
x x2 4 2
10.
7.
2. cot x cos ecx
2 32 32 3x 4 3x 1 27
1 sin 4 x 3 x 2sin 2 x 8 4
x 14. 2cot x 2
11.
8.
3. sec x tan x
x3 x 2 4 x 8log x 2 3
1 sin 6 x sin 2 x 2 6 2
15. log e x e x
16.
1 log a 2 sin 2 x b 2 cos 2 x 2 a b 2
20. log 1 e x
22.
sin 3 x 2sin 5 x sin 7 x 24. 3 5 7
27.
1 3 x log x 3
31.
1 x 1 log 4 3 x
28.
32.
x4 36. sin 1 5
3co s5 x 1 3 25. cos x cos x cos7 x 5 7
1 x4 tan 1 2 2
33.
x3 42. x 3log x 2 6 x 12 2 3 tan 1 3
49.
52.
46.
1 tan( x / 2) 2 3 log 3 tan( x / 2) 2 3 5 x 12 log 3cos x 2sin x 13 13
1 2 3
log
50.
19. log cos x sin x
41.
26.
1 1 5 x sin 5 3
34.
1 xn log n n x 1
2x2 1 38. sin 1 5
x3 37. sin 1 5
1 a tan x tan 1 ab b
29.
1 ex 1 log x 4 e 5
40. 2log sin 2 4sin 5 7 tan 1 sin 2
45.
sin x a sin x b
2 a log a b cos x 2 b a b cos x
1 4 sin x sin 2 x log sin x 4
1 3 1 cos 2 x cos 6 x 32 2 6
x3 1 1 x 4 tan 1 x 13. cos11x cos 3 x 3 22 6
18. cos ec a b .log
x 1
9.
5. tan x cot x 3 x
12.
17. sin a log sin x a x a cos a
21. 2log
x x2 4. 4 4
1 tan 2 tan 1 x 2
23.
1 tan 2 x log sec x 2
30.
x3 x 2 tan 1 x 3
35.
2 x2 1 1 tan 1 3 3
2 x3 39. log x 2 3 x 18 log 3 x6
1 1 log 2 x 2 6 x 5 tan 1 2 x 3 4 2
x 43. a sin 1 a 2 x 2 a
x 1 44. 3 5 2 x x 2 2sin 1 6
3 tan x 3 tan x
5 tan x 1 5 tan x 1
47.
1 2 5
2 tan x / 2 tan 1 3 3 2
53. x log x 2 x log x x
log
48. tan 1 tan 2 x
x 51. tan 1 1 tan 2 54. x sin 1 x 1 x 2
55. x cot
58.
x 2
1 1 sec x tan x log sec x tan x 2 2
x3 1 1 tan 1 x x 2 log x 2 1 3 6 6
61. e x cot
66.
56.
x 2
62.
59. 2 x tan 1 x log 1 x 2
x log x 1
63.
1 9 2x 7 2 x 7 7 x 10 x2 sin 1 4 8 3
ex 1 x
64.
67.
1 log x 2
x
57.
1 sin 1 x | log 1 x 2 2 1 x2
60.
1 1 x2 sin 1 x log x x
x log x
log x
65.
2
e ax a cos bx b sin bx a2 b2
16 8log log x
log x
2
16
68. x 2 x 13 2 7 x 1 x 2 x 1 3 log x 1 x 2 x 1 69. 1 1 x x 2 3 2 1 2 x 1 1 x x 2 5 sin 1 2 x 1 3 8 16 2 2 8 2 5
1 1 1 1 1 2 70. log x 1 1 cos x log x 2 log x 3 71. log 1 cos x log 1 cos x log 1 2cos x 6 3 2 2 6 3
72.
5 7 5 log x 2 log x 2 16 4 x 2 16
1 2 1 x x 73. log x 2 log x 2 4 tan 1 74. tan 1 x tan 1 2 2 3 3 2
1 1 1 2 tan 1 75. log 1 tan log tan 2 tan 1 tan 1 3 6 3 3
4
2 sin x 76. log 2 1 sin x
77.
x2 4 2 1 1 1 x2 2 x 1 1 x5 1 x2 x 1 1 x 1 78. log 2 79. tan 1 80. tan log log 5 5 x 1 2 x x 1 2 2 2 2 x 2 2 x 1 2 2 x 2x 4 2
81.
cot 1 tan 1 cot 2 cot 1 tan 2 tan 1 1 1 1 1 82. tan 1 log tan 1 log 2 cot 2 2 2 tan 2 2 2 cot 2 cot 1 2 tan 2 tan 1
2 2 83. 1 tan 1 tan x 1 84. 1 log x 2 x 1 85. 2 x tan 1 x 86. 1 log x 1 3 1 tan 1 x 1 2 2 x 2 2 x 1 2 4 3 x 1 3 2 2 tan x
87.
1 x2 x 1 1 1 1 88. 89. log x 2 1 tan 1 x log x 1 90. x cos 2 sin 2 .log sin x x 1 x 4 2 2
91.
ex x2
93.
a x2 x x 2 a 2 a log x
95.
x2 4 1 x 2 13 x 4 tan 1 log 8 3 x 2 13 x 4 3 x 16 3
96. log sec 2 x sec x tan x
98.
x sin log x cos log x 2
92. log x log cos x
1
ex 2 94. sin 1 3
97. 2sin e
99. 2log sin 2 4sin 5 7 tan 1 sin 2
100.
x
x2 2
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