INTEGRALS

EXERCISE – 1: Basic Subjective Questions

Section–A (1 Mark Questions) e2 x  1
19. Integrate with respect to x .
e2 x  1
1. Find  1  x  x dx. 20. Find the integral of sin 4 x with respect to x .
2 1
21. Find the integral of with respect to x
2. Evaluate  1  cos 2 xdx.  x  a  x  b 
5 x
 2x 
3. Find 4 e dx. 22. Integrate the function sin 1   with respect to x .
 1  x2 
 /4
4. Find 0 tan x dx. sin 4 x
 /2
1 dx
23. Evaluate 0 sin 4 x  cos4 x dx.
5. Evaluate 0 .
1  x2   1  sin x 
24. Evaluate  e x 
 /2  dx .
 1  sin x 
Section–B (2 Marks Questions)
Section–D (5 Marks Questions)
  cosec x  cot x  e
2 x
6. Evaluate dx.
2 dx
7. Evaluate 3
 sin x dx.
25. Evaluate the integral 0 x2  x  4.
1  cos 2 x 1
Evaluate dx. 26. Integrate the rational function with respect to x
8. 
1  cos 2 x x4  1
3
2  3sin x 2
9. Write the value of  dx. 27. Evaluate  (2 x  5 x) dx as limit of sum.
cos 2 x 1
2 cos x 3x  5
10. Evaluate  dx. 28. Integrate the rational function with
sin 2 x x3  x 2  x  1
0
1  tan x respect to x .
11. Write the value of  1  tan x dx . 5x  3
 29. Integrate the function with respect to x
2
4 x  4 x  10
12. Find  log x dx.
30. Integrate the function

x3 sin tan 1 x 4  with respect to
a 1  1  x8
13. If 0 dx  , then find the value of a.
4  x2 8 x.
Section–C (3 Marks Questions)
1 x
14. Evaluate 1 e dx.
 /2
15. Evaluate 0 cos 2 x dx.

sec2 x
16. Find the integral of with respect to x .
tan 2 x  4
17. Integrate e x sin x with respect to x .
1
18. Integrate with respect to x
1  cot x
EXERCISE – 2: Basic Objective Questions

Section–A (Single Choice Questions) dx
7. 02 1  sin x =
1 1
1. The value of  sin 2 x cos2 x dx is: (a) 0 (b)
2
(a) sin 2 x  cos 2 x  C (b) 1 3
(c) 1 (d)
(c) tan x  cot x  C (d) tan x  cot x  C 2
cos 2 x  cos 2 2 dx 2 dx
8. Let I1   and I 2   , then
2.  cos x  cos  dx is equal to : 1
1 x 2 1 x
(a) 2  sin x  x cos    C (a) I1  I 2 (b) I 2  I1
(b) 2  sin x  x cos    C (c) I1  I 2 (d) I1  2 I 2
(c) 2  sin x  2 x cos    C a
9. If a is such that 0 x dx  a  4, then
(d) 2  sin x  2 x cos    C
(a) 0  a  4 (b) 2  a  0
3. The value of (c) a  2 or a  4 (d) 2  a  4
sin x  cos x 3 7 d
 1  sin 2 x dx, 4  x  4 is equal to: 10. If  
3x 2  sin x  e x  6 x  cos x  e x , then the
dx
(a) log sin x  cos x  C (b) x  C representation of the expression in the form of anti-
derivative is
(c) log x  C (d)  x  C
(a) 3x  sin x  e x  C (b) 6 x  cos x  e  x  C
4. The value of  for which
3 x
(c) 6 x  cos x  e  x  C (d) 3x 2  sin x  e x  C
4x   4
 dx  log 4 x  x 4  C is
4x  x4
11. Evaluate:
x 3

 1  x  2
dx
(a) 1 (b) log e 4  2
x x2
(c) log 4 e (d) 4 x3 x 2 x 4 x3
(a)   xC (b)  C
3 2 4 3
1 1
5. If  dx  sin 1  x   C , then the value
23 x 4 x3 x4  x2  x
4  9x (c)   2 x2  C (d) C
4 3 4
of  is :
 e 
x log a
12. Find  ea log x  e a log a dx .
(a) 2 (b) 4 x a 1 xa
(a) a x   ax x  C (b) a x   xC
a 1 a
3 2
(c) (d)
2 3 ax x a 1
(c)   a a x  C (d) none of these
log e a a  1
y dt d2y 1
6. If x   and   y , then the value of
0
1  9t 2 dx 2
13. Evaluate:  4  9 x2 dx
 3x 
 is equal to : (a) tan 1 3x  C (b) tan 1    C
 2 
(a) 3 (b) 6 1  3x 
(c) tan 1    C (d) none of these
6  2 
(c) 9 (d) 1
 dx 
14. The value of  x2  4 x  8 is cos x  cos3 x dx is equal to:
20. 2
2
x 2
1  1  x2
(a) tan  C (b) tan 1  C 4 1
 2  2  2  (a) (b)
3 3
x 4 x3 1 8
(c)  C (d) x 2  x  C (c) (d)
4 3 6 5
dx 
15. The value of  5  8x  x2 is equal to: cos x dx is:
21. The value of 0
x4 (a) – 2 (b) 
(a) log C
x4 
(c) (d) 2
2
1 x4
(b) log C  3
2 21 x4 22. The value of  sin x cos 2 x dx is ______

1 21  x  4 (a) – 1 (b) 0
(c) log C
2 21 21  x  4 5 5
(c) (d)
(d) none of these 6 3
1

Evaluate:
dx 23. The value of 1 x x dx is:
16.  2
3  2x  x 
(a) 0 (b)
 x 1 2
(a) sin 1  C
 2  1
(c) 2 (d)
 x 2
(b) sin 1    C
2 

 x 1
24. Evaluate: 02 log  tan x  dx
(c) tan 1  C 1
 2  (a) 0 (b)
2
(d) tan 1  x   C 1
(c) 1 (d)
x 2
17. e sec x 1  tan x  dx  ________  C ?

 sin 
93
25. Evaluate: x  x 295 dx
(a) e x sec x (b) e x tan x

ex  
(c) (d) e x cos x (a) (b)
sec 2 x 2 2
4 4
18. If 1 f  x  dx  4 and 2  3  f  x   dx  7, then (c)  (d) 0
1
the value of 2 f  x  dx is 1 89
(a) 5 (b) – 1
26. Evaluate: 0 x(1  x) dx

(c) – 6 (d) – 5
1 1

(a) (b)
3
sin x  810 8190
19. If 02 3 sin x  3 cos x dx  a  2 , then a is equal to:

(a) 1 (b) 2 1 1
(c) (d)
792 910
3 1
(c) (d)
2 2
Case Study–3 Reason: If f is an odd function, then
30. The given integral can be transformed into a
 f ( x)dx
another form by changing the independent variable
 f  x  dx  0.
a
x to t by substituting x  g (t ) . (a) Assertion is correct, reason is correct; reason is a
Consider I   f ( x ) dx correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is
dx
Put x  g (t ) , so that  g '(t ) not a correct explanation for assertion.
dt (c) Assertion is correct, reason is incorrect.
We write dx  g '(t )dt (d) Assertion is incorrect, reason is correct.
Thus, I   f ( x ) dx   f  g (t )  g '(t ) dt . 33. Assertion: If the derivative of function x is
Based on the above information, answer the d
 x   1, then its anti-derivatives or integral is
following questions. dx
(i) 2
 1)dx  ______ . d  x n 1 
 2 x sin( x  dx  x  C . Reason: If 
dx  n  1 
n
  x , then the
(a) cos( x 2  1)  c (b)  cos( x 2  1)  c
corresponding integral of the function is
(c) 2 cos( x 2  1)  c (d) 2 cos( x 2  1)  c n x n 1
x dx 
n 1
 C , n  1.
sin(tan 1 x) (a) Assertion is correct, reason is correct; reason is a
(ii)  1  x 2 dx is equal to correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is
(a)  sin(tan 1 x)  c (b)  cos(tan 1 x)  c not a correct explanation for assertion.
(c) tan x  c (d) None of these (c) Assertion is correct, reason is incorrect.
(d) Assertion is incorrect, reason is correct.
(iii)  tan xdx is equal to 34. Assertion: The value of the integral
(a) sec x  c x 2 x
(b) cot x  c  e  tan x  sec x  dx is e tan x  C
(c) log | x |  c (d) None of these Reason: The value of the integral
2x e x  f  x   f '  x  dx is e x f  x   C.
(iv)  1  x 2 dx is equal to (a) Assertion is correct, reason is correct; reason is a
2 2
correct explanation for assertion.
(a) 1  x  c (b) log 1  x  c (b) Assertion is correct, reason is correct; reason is
not a correct explanation for assertion.
2 (c) Assertion is correct, reason is incorrect.
(c) log c (d) None of these
1  x2 (d) Assertion is incorrect, reason is correct.
b b
Section–C (Assertion & Reason Type Questions) 35. Assertion: The value of  f  t  dt and  f  u du are
a a
 
 equal
31. Assertion:  x sin x cos2 x dx   sin x cos
2
x dx
0
2 0 Reason: The value of definite integral of a function
b b over any particular interval depends on the function
ab
Reason:  xf  x  dx  2 a
f  x  dx and the interval and not on the variable of
a integration.
(a) Assertion is correct, reason is correct; reason is a (a) Assertion is correct, reason is correct; reason is a
correct explanation for assertion. correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is (b) Assertion is correct, reason is correct; reason is
not a correct explanation for assertion. not a correct explanation for assertion.
(c) Assertion is correct, reason is incorrect. (c) Assertion is correct, reason is incorrect.
(d) Assertion is incorrect, reason is correct. (d) Assertion is incorrect, reason is correct.
2
1 x 
32. Assertion:  log  1  x  dx  0.
2
EXERCISE – 3: Previous Year Questions
 1  x3
1. Find the antiderivative of  3 x   . (Delhi 2014) 19. Find:  x 4  3x 2  2 dx (AI 2014C)
 x
1 3x  1
2. Evaluate:  cos  sin x  dx (Delhi 2014) 20. Evaluate:   x  12  x  3 dx (Delhi 2013C)
dx
3. Evaluate:  sin 2 x cos2 x (Foreign 2014)
3x  5
21. Evaluate:  x3  x2  x  1 dx (Delhi 2013C)
4. Evaluate:  1  x  xdx (Delhi 2012)
8
3 22. Evaluate:   x  2 dx (AI 2013C)
5. Write the value of   ax  b  dx (AI 2011) x 2
4 
x3  x 2  x  1 23. Evaluate:
2
dx (Delhi 2012)
6. Evaluate:  dx (Delhi 2012)  1  x  1  x 2
x 1  
sin 6 x 2 x dx
7. Find:  cos8 x dx (AI 2014C) 24. Evaluate:  (Delhi 2011)
x 2

 1 x2  3 
x  cos 6 x
8. Write the value of  3x 2  sin 6 x dx (AI 2012C)
25. Given x x
 e  tan x  1 sec xdx  e f  x   c Write

 3sin   2  cos  f  x  satisfying above. (AI 2012)

9. Find  5  cos2   4 sin  d . (Delhi 2016, 2013C)

26. Find:   3x  1 4  3 x  2 x 2 dx (AI 2016)

sin  x  a 
10. Evaluate:  dx (Foreign 2015)
sin  x  a 
27. Find:   x  3 3  4 x  x 2 dx. (Delhi 2015, 20114C)
dx
11. Write the value of  x 2  16 . (Delhi 2011)
28. Integrate w.r.t. x.
x 2  3x  1
(Delhi 2015)
1  x2
x
12. Find  dx. (Delhi 2016)
a  x3
3
Find:  
x 2  1 log x 2  1  2 log x 
 dx 
sin x  cos x
29.  x4
13. Evaluate:  dx (Delhi 2011C)
sin 2 x (AI 2014C, 2012C)
1 sin 1 x  cos 1 x
14. Evaluate:  cos4 x  sin 4 x dx (AI 2014) 30. Find:  sin 1 x  cos 1 x
dx, x   0,1 (AI 2014C)

1 5
15. Evaluate:  sin 4 x  sin 2 x cos2 x  cos4 x dx 31. Evaluate: x
2

 3 dx as limit of sums.
2
(AI 2014)
(Delhi 2012C)
6x  7
16. Evaluate:   x  5 x  4  dx (AI 2011) 4

x 
2
32. Evaluate:  x dx as limit of sums.
1
2
x
17. Find:  x4  x 2  2 dx (AI 2016) (AI 2012C, Delhi 2010)
2
x   dx  3x 
2 2
 1 x2  4 33. Evaluate:  2 dx as limit of sums.
18. Find:  (Foreign 2016) 0
x 2
 3 x 2
 5
(AI 2011C)
 1  /2
1 dx
34. Evaluate:  dx (AI 2014C, 2011C) 50. Evaluate:  (Delhi 2015C)
2 1  tan x
0 1 x 0

2 4
x3  1
35. Evaluate:  dx (Delhi 2011) 51. Evaluate: 0  x  x  2  x  4  dx (Delhi 2013)
1
x2
5
3
1 52. Evaluate:   x  2  x  3  x  5  dx (Delhi 2013)
36. Evaluate:  dx (Delhi 2012) 2
2
x
2
1
2
4  x dx 2
53. Evaluate:  1  esin x dx (Delhi 2013)
37. Evaluate:  (AI 2012) 0
0

x
38. Evaluate: 
1
dx
(Delhi 2011C)
54. Evaluate:  1  sin x dx (AI 2012C, Delhi 2010)
2 0
0 1 x
1
1 
 /2
2
x sin x dx
55. Evaluate:  log  x  1 dx (AI 2011)
39. Evaluate:  (Delhi 2014C) 0
0

1 4 3
x 1 1
56. Evaluate:  1 dx (AI 2011, Delhi 2007)
40. Evaluate:  x2  1 dx (AI 2011C)
 tan x
0
6
4 x
41. Evaluate: 2 2
dx (AI 2011C) 4
x 1 Evaluate:
57.   x  1  x  2  x  4 dx (Delhi 2011C)
1
e2
dx
42. Evaluate:  . (AI 2014) 2 x3
x log x 58. x e dx equals (AI 2020, Delhi 2020)
e

1 4
2
x 59. Find the value of  x  5 dx. (AI 2020, Delhi 2020)
43. Evaluate:  xe dx (Foreign 2014)
1
0

1 x
tan 1 x 60. Find  x 2  3x  2 dx. (AI 2020)
44. Evaluate:  2
dx (AI 2014C)
0 1 x
2 1 1  2x
ex 1
61. Evaluate 1  x  2 x2  e dx (Delhi 2020)
45. Write the value of  dx (Delhi 2012 C)
0 1  e2 x
1
n
Evaluate:
 /4 sin 2 d 62. Find the value of  x 1  x  dx. (Delhi 2020)
46. 0 sin 4   cos4 
(AI 2013C) 0

 /4 e x 1  x  dx
sin x  cos x 63.  is equal to (AI 2020)
47. Evaluate:  dx (Delhi 2014C)
0
9  16 sin 2 x  
cos 2 xe x

 /4
 64. Evaluate:  4 x 3 x dx (Delhi 2020)
48. Evaluate:  
tan x  cot x dx  2.
2
0 dx
65. Find:  9  4 x2 (AI 2020)
(Delhi 2012)
2  
3 66. Find:  1  sin 2 xdx, x (Delhi 2019)
49. Evaluate:  x  x dx. (Delhi 2016, AI 2012, 2011) 4 2
1
 cos 2 x  2 sin 2 x
67. Evaluate:  dx (2018)
cos 2 x
cos( x  a )
68. Integrate with respect to x (AI 2019)
sin( x  b)

sec 2 x
69. Find:  dx (Delhi 2019)
tan 2 x  4
dx
70. Find:  (AI 2019)
5  4 x  2 x2
3x  5
71. Find:  2
dx (Delhi 2019)
x  3x  18

72. Find:  sin 1 (2 x) dx (Delhi 2019)

73. Find:  x  tan 1 xdx (AI 2019)

0 1  tan x
74. Find:   1  tan x dx (AI 2019)
4

75. Evaluate 10 x2 (1  x)n dx (AI 2019)

 Answer Key

EXERCISE-1:
Basic Subjective Questions

3 5
2 2 2 2 3x 1 1
1. x  x C 2. tan x  C 20.  sin 2 x  sin 4 x  C
3 5 8 4 32
3. e4  e  1 4. log 2   a  b 
21. log  x      x  a  x  b  C
 x
  2 
5. 6. e cot x  C
2 22. 2 x tan 1 x  log 1  x 2   C
1 cos 3x  
7.  3cos x  C 8. log sec x  C 
4 3  23. 24. e 2
4
9. 2 tan x  3sec x  C 10. 2cosec x  C
1  5  15 1  15 
1 25.  log  
11. log 2 12. x log x  x  C 5  5  15 1  15 
2
e2  1 1 1 1
13. a  2 14. 26.  log x  1  log x  1  tan 1 x  C
2 4 4 2
15. 0 112
27.
3
16. log tan x  tan 2 x  4  C
1 x 1 4
28. log  C
ex 2 x  1  x  1
17.  sin x  cos x   C
2
29. 5 x 2  4 x  10  7 log  x  2   x 2  4 x  10  C
x 1
18.  log sin x  cos x  C
2 2 1
x x 4

30.  cos tan 1 x 4  C
19. log e  e C
 EXERCISE-2:
Basic Objective Questions

1. (d) 2. (a) 3. (d) 4. (b) 5. (c)

6. (c) 7. (c) 8. (b) 9. (d) 10. (d)

11. (a) 12. (c) 13. (c) 14. (b) 15. (c)

16. (a) 17. (a) 18. (d) 19. (d) 20. (a)

21. (d) 22. (b) 23. (a) 24. (a) 25. (d)

26. (b) 27. (a)

28. (i) (a) (ii) (b) (iii) (c) (iv) (c)

29. (i) (b) (ii) (b) (iii) (a) (iv) (d)

30. (i) (b) (ii) (b) (iii) (d) (iv) (b)

31. (c) 32. (a) 33. (a) 34. (a) 35. (a)
EXERCISE-3:
Previous Year Questions

 x2 1  x  27 x 5
1. 2 x  x  1  C 2. x C 18. x  tan 1   log C
2 2 4 3  3  8 5 x  5
3 5
2 2 2 2 1
3. tan x  cot x  C 4.
3
x  x C
5 19.
2

2 log x 2  2  log x 2  1  C
4

5.
 ax  b  C 6.
1 3
x  xC
x 1 1
20. 2 log  C
4a 3 x  3 x 1
1 7 1 x 1 4
7. tan x  C 21. log  C
7 2 x 1 x 1
1
8. log  3x 2  sin 6 x   C  x2  1  x 
6 22. log    tan    2
4
2
 x 4 2
9. 3log  sin   2   C
sin   2 1
 
23.  log  x  1  log 1  x 2  tan 1 x  C
2
10. x cos 2a  sin 2a log sin  x  a   C
1 x2  1
3
24. log 2 C 25. f  x   sec x
1  x 2 1  x  2
2 x 3
11. tan 1    C 12. sin    C
4 4 3 a 1 2
1 5  3   41   3
13.  log sin x  cos x  sin 2 x  C 26. 
2
 4  3x  2 x 2  2   x      x  
4 2 4   16   4
1  tan x  cot x  205  4x  3 
14. tan 1  C  sin 1  C
2  2  64 2  41 
1  tan x  cot x  1 2
 x  2 3  4x  x2
15. tan 1  C
3  3 
27. 
3
 3  4x  x2 3 
2
 9 7  x2
16. 6 x 2  9 x  20  34 log  x    x 2  9 x  20  C  sin 1  C
 2 2  7 
1 x 1 2  x  x 3
17. log  tan 1  C 28. 1  x 2  sin 1 x  1  x 2  C
6 x 1 3  2 2 2
  3 3
  23
1  x 2  1  2  x2  1  2  x2  1  2  56. 57.
29.   log  2     C 12 2
3  x 2   x  3  x2  
  1 3 15
58. e x  C 59.
4 2 3 2
30. x 

x cos 1 x 

 sin 1
x  x 1 x  C x 1
60. log x 2  3x  2  3log C
27 x2
31. 48 32.
2 e2  e2  2  1
 61. 62.
33. 4 34. 4  n  1 n  2 
2
12 x
3 63. tan  xe x   C 64. C
35. 1 36. log   log12
2
 1  2x 
37.  38. 65. tan 1    C 66.   cos x  sin x   C
4 6  3 
3  4 67. tan x  C
39.   2 40.
6 68. cos  a  b  log sin  x  b   x sin  a  b   C
1  17 
41. log   42. log 2 69. log tan x  tan 2 x  4  C
2  5
1 2 1  2 
43.  e  1 44. 70. sin 1   x  1   C
2 32 2  7 
 13 14
45. tan 1 e  tan 1 1 46. 71. log  x  6   log  x  3  C
4 9 9
1 11 1 1
47. log 3 49. 72. x sin  2 x   1  4 x2  C
20 4 2
 1 x 1
50. 51. 20 73. 1  x 2  tan 1 x   C 74. log 2
4 2 2 2
23 1 1 2
52. 53.  75.  
2  n  1  n  3 n2
54.  55. 0