CLASS 12 DEFINITE INTEGRATION DPP
DEFINITE INTEGRAL
✤ PRACITCE QUESTION
Very Short (Objective Type) / Short Answer Type
1. If a is such that ∫₀ª x dx ≤ a+ 4, then
(a) 0 ≤ a ≤ 4 (b) -2 ≤ a ≤ 0
(c) a ≤ -2 or a ≤ 4 (d) -2 ≤ a ≤ 4
2. If ∫₁⁴ f(x) dx = 4 and ∫₂⁴ (3 - f(x)) dx = 7, then the value of ∫₁² f(x)dx is
_____.
Evaluate the following integrals in Exercises 3 to 7:
3. ∫₀¹ (2x)/(1 + x²) dx
4. ∫₂³ (1/x) dx [Delhi 2012]
5. ∫₀³ (dx)/(9 + x²) [Delhi 2014]
6. ∫₀π/2 cos 3x cos 2x dx
7. ∫₀¹ (eˣ)/(1 + e²ˣ) dx
8. If ∫₀¹ (3x² + 2x + k)dx = 0, find the value of k.
Long Answer I / Long Answer II Type
Evaluate the following integrals in Exercises 9 to 22:
9. ∫₀π/4 (sin x + cos x)/(9 + 16 sin 2x) dx [NCERT; DoE]
10. ∫₁² (1)/(x(1 + log x)²) dx [Foreign 2014, 2011]
11. ∫₀ᵃ √(a² - x²) dx
12. ∫₀¹ x sin⁻¹ x dx
13. ∫₀¹ x/(1 + x⁴) dx
14. ∫₀π/4 1/(1 + tan x) dx
15. ∫₀¹ tan⁻¹(2x/(1 - x²)) dx
16. ∫₀¹ x log(1 + 2x) dx
17. ∫₀π/2 eˣ((1 - sin x)/(1 - cos x)) dx
18. ∫₀π/4 sin x cos x/(cos² x + sin⁴ x) dx
19. ∫₀π/2 sin 2x tan⁻¹(sin x) dx
20. ∫₀¹ dx/(eˣ + e⁻ˣ)
21. ∫₀π/4 dx/(cos³ x √2 sin 2x)
22. ∫₀π/2 dx/(a² cos² x + b² sin² x)
❈ VERY SHORT (OBJECTIVE TYPE) / SHORT ANSWER TYPE QUESTION
1. ∫-11(x³/x³+1) dx is equal to ___________.
2. ∫a2af(x)dx = 2f(x)dx if f(2a-x)=f(x). State true or false.
3. ∫π/20(∛sin x / ∛sin x + ∛cos x) dx = a * π/2, then value of a is ___.
4. Evaluate ∫ π/2 -π/2sin5x dx
5. Evaluate ∫1/2 -1/2 cos x log (1+x/1-x)dx
6. Evaluate ∫1-1 x³ /(1+x²) dx [HOTS]
7. Evaluate ∫-π/4π/4 x3 sin4 x dx
8. Evaluate ∫-1 1 |x|dx
9. Evaluate: ∫0 to 2π cos⁵x dx
10. Evaluate ∫0 to π/2 log(tan x) dx
11. Evaluate ∫0 to 1 [2x] dx
12. Evaluate ∫1 to 4 f(x) dx, where
f(x) = {2x+8, 1 ≤ x ≤ 2; 6x, 2 ≤ x ≤ 4}
13. Evaluate ∫-1 to 1 log |(2+3x) / (2-3x)| dx
Long Answer I / Long Answer II Type
14. Evaluate ∫0 π (e(cos x) / (e(cos x) + e-cos x)) dx
15. Evaluate ∫0 to 1 cot⁻¹(1-x+x²) dx
16. Evaluate ∫-1 to 2 |x³ - x| dx [NCERT; Delhi, 2016; AI 2012]
17. Evaluate ∫0 to π (x tan x) / (sec x cosec x) dx
18. Evaluate ∫0 to π/2 log (sin x) dx
19. Evaluate ∫0 to π (x / (a² cos² x + b² sin² x)) dx
20. Evaluate ∫₀¹ log(1+x) / (1+x²) dx
21. Evaluate ∫₀π x tan x / (sec x + tan x) dx.
22. Evaluate ∫₀π/2 (2 log sin x - log sin 2x) dx [NCERT: DoE]
23. Evaluate ∫₀¹ log((1-x)/x) dx OR ∫₀¹ log(1/x - 1) dx
24. Evaluate ∫₀²ˣ 1 / (1+esin x) dx
25. Prove that ∫₀π/4 log(1+tan θ) dθ = (π/8) log 2
26. Evaluate ∫₀² |x² + 2x - 3| dx
27. Evaluate ∫₀¹ log x / √(1-x²) dx
28. Evaluate: ∫₀-3/2 |x sin πx| dx
29. Evaluate ∫π/3 π/6 dx / (1 + √cot x)
30. Evaluate ∫₁³ |x³ - x| dx
31. Evaluate ∫₀π/2 x sin x cos x / (sin⁴ x + cos⁴ x) dx
36. Prove that ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a - x) dx and hence, prove that
∫₀π/2 (sin x / (sin x + cos x)) dx = π/4
❉ PRACTICE QUESTION
✤ Very Short (Objective Type) / Short Answer Type
1. If ∫₀¹ (1 / (1 + 4x²)) dx = π/8, then a = ____.
2. ∫₋₁⁰ |(1 - x)| dx is equal to ____.
3. ∫₋π/2π/2 (√cos x - cos³x) dx is equal to 0. State true or false.
4. The value of ∫₀^(π) |cos x| dx is 2. State true or false.
5. The value of ∫₋π^(π) sin³ x cos²x dx is ____.
Evaluate the following integrals in Exercises 6 to 15:
6. ∫₋ππ (sin -93x + x²⁹⁵) dx
7. ∫₋π/2π/2 log |(2 - sin x) / (2 + sin x)| dx
8. ∫₋π/4π/4 |sin x| dx [DoE]
9. ∫0π/2 ((sin x - cos x) / (1 + sin x cos x)) dx
10. ∫₁e log x . dx
11. ∫₀¹ x(1 - x)⁸⁹ dx
12. ∫₀ᵃ (f(x) / (f(x) + f(a - x)) dx [Dehradun 2019]
13. ∫₀¹ x²(1 - x)ⁿ dx
14. ∫₁³ √(4 - x) / (√x + √(4 - x)) dx [DoE]
15. ∫₀π |cos x| dx [DoE]
✤ Long Answer 1 / Long Answer II Type
Evaluate the following integrals in Exercises 16 to 28:
16. ∫₀^(3/2) |x cos πx| dx [AI 2016; Patna 2015]
17. ∫₁⁴ (|x - 1| + |x - 2| + |x - 4|) dx
18. ∫₋₅⁵ |x + 2| dx
19. ∫₋₁^(3/2) |x sin πx| dx [NCERT; DoE]
20. ∫₋ππ (cos ax - sin bx)² dx [Delhi 2015]
21. ∫₀π/2 x cot x dx [HOTS]
22. ∫π/6π/3 (1 / (1 + √tan x)) dx [DoE; HOTS]
23. ∫₀⁸ |x - 5| dx
24. ∫28 3√x+1 / 3√x+1 + 3√11-x dx
25. ∫−π/2π/2 [sin|x| - cos|x|]dx [DoE]
26. ∫0π/2 √cot x / √cot x + tan x dx [NCERT]
27. ∫0π x sin x / 1 + cos2 x dx
[NCERT; DoE; Delhi 2017(C), AI 2013, 12]
28. ∫0π x / 1 + sin x dx [NCERT; NCERT Exemplar; DoE]
❉ INTEGRATED EXERCISE
✤ Very Short (Objective Type) / Short Answer Type
1. ∫(1/x²(x⁴+1)³/⁴) dx is equal to
(a) -(1 + 1/x⁴)¹/⁴ + C
(b) (x⁴+1)³/⁴ + C
(c) (1 - 1/x⁴)¹/⁴ + C
(d) -(1 + 1/x⁴)³/⁴ + C
2. ∫xeˣ/(1+x)² dx is equal to
(a) eˣ/x+1 + C
(b) eˣ(x+1) + C
(c) -eˣ/(x+1)² + C
(d) eˣ/1+x² + C
3. If ∫2ˣ/√(1-4ˣ) dx = p sin⁻¹(2ˣ) + C, then 'p' is equal to
(a) logₑ2
(b) 1/2 logₑ2
(c) 1/2
(d) 1/logₑ2
4. The value of integral ∫(π/4,0) (sin x + cos x)/(9+16 sin 2x) dx is
(a) log 2
(b) 1/20 log 2
(c) 1/20 log 3
(d) log 5
5. The value of integral ∫(1/2,1) cos x.log((1+x)/(1-x)) dx is
(a) 0 (b) ½ (c) 3/2 (d) none of these
6. If [x] denotes the greatest integer less than or equal to x, then the
value of the integral ∫(2,0) x²[x]dx is _________ .
7. ∫31 √x / [√(6-x)+√x] dx is equal to 2. State true or false.
Evaluate the following in Exercises 8 to 19:
8. ∫₀π/2 sin 2x dx [Foreign 2014]
9. ∫ ex / (1 + e^(2x)) dx
10. ∫ x / √x+2 dx
11. ∫ 1 / (cos²x(1 - tan x)²) dx
12. ∫ tan⁻¹(cot x) dx [HOTS]
13. ∫ cos 3x cos x dx
14. ∫ sin 3x sin 2x dx
15. ∫₂⁴ x / x² + 1 dx [AI 2014]
16. ∫ e² / x log x dx [AI 2014]
17. ∫₀π (sin² x/2 - cos² x/2) dx [NCERT]
18. ∫₋ππ x¹⁰ sin⁷x dx
19. ∫ {sin(log x) + cos(log x)} dx
20. Given ∫eˣ(tan x + 1) sec x dx = eˣ f(x) + C. Write the value of f(x).
[AI 2012]
21. If ∫((x-1)/x²)eˣ dx = f(x)eˣ + C, then write the value of f(x).
[Foreign 2014]
22. If ∫₀ᵃ 3 x²dx = 8 write the value of a. [Foreign 2014]
✤ Long Answer I / Long Answer II Type
Evaluate the following in Exercises 23 to 83:
23. ∫ dx / (√2x+1 + √2x+2) dx
24. ∫ 1 / (sin(x-p)cos(x-q)) dx
25. ∫ sin x / sin(x-a) dx
26. ∫ (x+3) √3-4x-x² dx [Delhi 2015]
27. ∫ dx / (sin x + sin 2x) [Delhi 2015]
28. ∫ (1-cos x) / (cos x (1+cos x)) dx [Chennai 2015]
29. ∫ x(log x)² dx [NCERT]
30. ∫ (sin⁸ x - cos⁸ x) / (1-2 sin² x cos² x) dx [NCERT]
31. ∫ (sin⁶ x + cos⁶ x) / (sin² x cos² x) dx
[NCERT Exemplar; DoE; Delhi 2014]
32. ∫ sin³ x cos³ x dx
33. ∫ dx / x(x⁵ + 2)
34. ∫ 1 / x(x⁴ + 1) dx [NCERT; HOTS]
35. ∫₀π/2 (sin x + cos x) / (3 + sin 2x) dx [Ajmer 2015]
36. ∫ ex (sin⁻¹ x + 1 / √1-x²) dx
37. ∫ log (1+x²) dx
38. ∫ x tan⁻¹ x dx [NCERT]
39. ∫ 1 / (cos(x+a) cos(x+b)) dx
40. ∫ (x² + 1) / ((x² + 4)(x² + 25)) dx [Delhi 2013]
41. ∫ ex(sec x + sec x tan x) dx [NCERT; HOTS]
42. ∫ (x²-3x+1) / √1-x² dx [Delhi 2015]
43. ∫ 2x / ((1+x²)(3+x²)) dx [NCERT]
44. ∫ 1 / (1 + cot x) dx [NCERT]
45. ∫ 1 / x³(x⁴+1)³/⁴ dx
46. ∫ tan⁵x sec⁴x dx [NCERT Exemplar]
47. ∫ cos 2x / (cos x + sin x)² dx [NCERT]
48. ∫₀π/2 cos² x / (1 + 3 sin² x) dx [Ajmer 2015]
49. ∫-π/2 cos x / (1 + e^x) dx
50. ∫√(a-x)/(a+x) dx
51. ∫√cot x dx
52. ∫ex(log x + 1/x) dx
53. ∫[log(log x) + 1/(log x) 2] dx [NCERT; Bhubaneswar]
54. ∫sin-1(2x/(1 + x2)) dx
[NCERT]
55. ∫1/(cos4 x + sin4 x) dx
[AIMS]
56. ∫0π/2 sin 2θ/(sin4 θ + cos4 θ) dθ
[IIT]
57. ∫0π/2 √sin φ cos5 φ d φ.
[NCERT; HOTS]
58. ∫1/(sin4 x + sin2 x cos2 x + cos4 x) dx
[AIIMS]
59. ∫x cos-1 x / √1 - x2 dx
[MET]
60. ∫log x/(x + 1)2 dx
[AIMS]
61. ∫(3x - 2)√(x2 + x + 1) dx
[Foreign 2015]
62. ∫e2x sin(3x + 1) dx
[Foreign 2015]
63. ∫(√cot x + √tan x) dx
[NCERT: Patna 2015]
64. ∫(x3 - 1)/(x3 + x) dx
[Patna 2015]
65. ∫cos-1 (sin x) dx
[Delhi 2014]
66. ∫0π/2 ex (sin x - cos x) dx
[Delhi 2014]
67. ∫0π/2 ex / (x log x) dx
[AI 2014]
68. ∫0+ 2sin x/(2sin x + 2cos x) dx
[Patna 2015]
69. ∫01 x e^(x^2) dx
[NCERT; Foreign 2014]
70. ∫-12 f(x) dx, where f(x) = |x + 1| + |x| + |x - 1|
[NCERT Example]
71. ∫ √(1+x²) dx [Panchkula 2015]
72. ∫(x+3)eˣ / (x+5)² dx [NCERT Exemplar]
73. ∫₀π x sin x cos²x dx [NCERT]
74. ∫₀π/2 sin³/² x / sin³/² x + cos³/² x dx [NCERT]
75. ∫₀1 [x]dx [NCERT: HOTS]
76. ∫₀1 x(1-x)n dx
77. ∫₀π/4 sec x / 1 + 2 sin²x dx [Guwahati 2015]
78. ∫₀π/4 (1 - sin 2x) / 1 - cos 2x dx [Guwahati 2015]
79. ∫₀π/2 5 sin x + 3 cos x / sin x + cos x dx [Bhubneshwar 2015]
80. ∫ x / 1 + tan x dx [Bhubneshwar 2015]
81. ∫ sin x - x cos x / x(x + sin x) dx [Ajmer 2015]
82. ∫₀π sin²x / sin x + cos x dx
[NCERT Exemplar; Al 2016; Panchkula 2015]
83. ∫ cosec³x dx [HOTS]
84. Evaluate ∫₁³ (2x² + x + 9) dx as the limit of a sum. [HOTS]
85. Evaluate ∫₀² (x² + e^(2x) + 1) dx as the limit of a sum.
[Chennai 2015]
86. ∫₋₁² (e³x + 7x - 5) dx as a limit of sums. [Panchkula 2015]
87. Evaluate ∫₁³ (e^(-3x) + x² + 1) dx as a limit of sums. [Delhi 2015]
88. Prove that ∫₀¹ √1 - x / 1 + x dx = π/2 - 1.
89. If ∫₀ᵃ 1 / 4 + x² dx = π/8, find the value of a. [AI 2014]
90. Prove that ∫₀^π x sin x / 1 + sin x dx = π(π - 2).
* * * * * * * * * * * * * *
