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๐Ÿ“˜ DEFINITE INTEGRATION โ€“ FROM CONCEPTS TO JEE ADVANCED (100 QUESTIONS by Grandmaster Bikram Sutradhar)

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๐Ÿ“˜ DEFINITE INTEGRATION โ€“ FROM CONCEPTS TO JEE ADVANCED (100 QUESTIONS by Grandmaster Bikram Sutradhar)

๐Ÿ“˜ DEFINITE INTEGRATION โ€“ FROM CONCEPTS TO JEE ADVANCED (100 QUESTIONS by Grandmaster Bikram Sutradhar)

DEFINITE INTEGRATION DEFINITE INTEGRATION

๐Ÿ“˜ DEFINITE INTEGRATION โ€“ FROM CONCEPTS TO JEE ADVANCED

100 Questions โ€ข JEE Main & Advanced

By GrandMaster Bikram Sutradhar

โœฆ Introduction

Definite Integration is the backbone of area under curve, properties of integrals, and advanced substitution techniques. In JEE-level examinations, questions are rarely direct; instead, they rely heavily on symmetry, periodicity, and property-based transformations.

This article presents hard definite integrals derived from standard indefinite forms but transformed using competitive limits such as:

  • \([0,1]\)
  • \([0,\pi/2]\)
  • \([-\pi/2,\pi/2]\)
  • \([0,\infty)\)
  • Symmetric limits \([-a,a]\)

๐Ÿ”น Important Properties Used

  • \(\int_{-a}^{a} f(x)\,dx = 0\) if \(f(x)\) is odd
  • \(\int_{0}^{a} f(x)\,dx = \int_{0}^{a} f(a-x)\,dx\)
  • \(\int_{0}^{\pi/2} f(\sin x, \cos x)\,dx\) (symmetry property)
  • Periodic integrals
  • Substitution: \(x \to a-x\), \(x \to \pi/2-x\)

๐Ÿ“— DEFINITE INTEGRATION โ€“ QUESTION SET

A. Algebraic & Rational (Definite Form)

1. \( \int_{0}^{1} \frac{x^3+1}{x^2+1}\,dx \)

2. \( \int_{0}^{1} \frac{x^4+1}{x^2+1}\,dx \)

3. \( \int_{0}^{1} \frac{x^2}{(x^2+1)^2}\,dx \)

4. \( \int_{0}^{1} \frac{x^3}{(x^2+1)^2}\,dx \)

5. \( \int_{0}^{1} \frac{x^5}{x^2+1}\,dx \)

6. \( \int_{0}^{1} \frac{x}{x^4+1}\,dx \)

7. \( \int_{0}^{1} \frac{x^2}{x^4+1}\,dx \)

8. \( \int_{0}^{1} \frac{x^3}{x^4+1}\,dx \)

9. \( \int_{0}^{\infty} \frac{dx}{x^4+4} \)

10. \( \int_{0}^{\infty} \frac{dx}{x^4+1} \)

B. Trigonometric (JEE Favourite)

11. \( \int_{0}^{\pi/2} \frac{\sin x}{1+\cos x}\,dx \)

12. \( \int_{0}^{\pi/2} \frac{\cos x}{1+\sin x}\,dx \)

13. \( \int_{0}^{\pi/2} \frac{dx}{\sin x+\cos x} \)

14. \( \int_{0}^{\pi/2} \frac{dx}{\sin x-\cos x} \)

15. \( \int_{0}^{\pi/2} \tan^3 x \sec^2 x\,dx \)

16. \( \int_{0}^{\pi/2} \sin^5 x \cos^3 x\,dx \)

17. \( \int_{0}^{\pi/2} \frac{dx}{\sin^3 x} \)

18. \( \int_{0}^{\pi/2} \frac{dx}{\cos^3 x} \)

19. \( \int_{0}^{\pi/2} \sec x \tan^2 x\,dx \)

20. \( \int_{0}^{\pi/2} \csc x \cot^2 x\,dx \)

C. Exponential & Logarithmic

21. \( \int_{0}^{1} x e^{x^2}\,dx \)

22. \( \int_{0}^{1} x^2 e^x\,dx \)

23. \( \int_{0}^{\pi/2} e^x \sin x\,dx \)

24. \( \int_{0}^{\pi/2} e^x \cos x\,dx \)

25. \( \int_{0}^{1} x e^x \cos x\,dx \)

26. \( \int_{0}^{1} \ln(x^2+1)\,dx \)

27. \( \int_{0}^{1} x\ln x\,dx \)

28. \( \int_{1}^{e} \ln(\ln x)\,dx \)

29. \( \int_{1}^{e} \frac{\ln x}{x^2}\,dx \)

30. \( \int_{1}^{e} \frac{\ln x}{x}\,dx \)

D. Radical & Special Forms

31. \( \int_{0}^{1} \sqrt{x^2+1}\,dx \)

32. \( \int_{1}^{2} \sqrt{x^2-1}\,dx \)

33. \( \int_{0}^{1} \frac{dx}{\sqrt{x^2+1}} \)

34. \( \int_{1}^{2} \frac{dx}{\sqrt{x^2-1}} \)

35. \( \int_{0}^{1} \frac{x}{\sqrt{x^2+1}}\,dx \)

36. \( \int_{0}^{1} \frac{x^2}{\sqrt{x^2+1}}\,dx \)

37. \( \int_{1}^{2} \frac{dx}{x\sqrt{x^2-1}} \)

38. \( \int_{0}^{1} \frac{dx}{x\sqrt{1-x^2}} \)

39. \( \int_{0}^{\pi/2} \sqrt{1+\sin x}\,dx \)

40. \( \int_{0}^{\pi/2} \sqrt{1-\cos x}\,dx \)

E. Symmetry & Property-Based (Very Important)

41. \( \int_{-1}^{1} \frac{dx}{x^2\sqrt{x^2-1}} \)

42. \( \int_{-a}^{a} \frac{\sqrt{x^2-1}}{x}\,dx \)

43. \( \int_{0}^{1} \frac{x}{\sqrt{x^2+x+1}}\,dx \)

44. \( \int_{0}^{1} \sqrt{x^2+x+1}\,dx \)

45. \( \int_{0}^{1} \frac{dx}{\sqrt{x^2+x+1}} \)

46. \( \int_{0}^{1} \frac{dx}{x^2+x+1} \)

47. \( \int_{0}^{1} \frac{x^2}{x^2+x+1}\,dx \)

48. \( \int_{0}^{1} \frac{dx}{\sqrt{ax^2+bx+c}} \)

49. \( \int_{0}^{1} \sqrt{ax^2+bx+c}\,dx \)

50. \( \int_{0}^{1} \frac{x}{\sqrt{ax+b}}\,dx \)

Author: GrandMaster Bikram Sutradhar
Institute: bAstronautWay
Level: JEE Main & Advanced

๐Ÿ“˜ DEFINITE INTEGRATION โ€“ FROM CONCEPTS TO JEE ADVANCED (100 QUESTIONS by Grandmaster Bikram Sutradhar)
๐Ÿ“˜ DEFINITE INTEGRATION โ€“ FROM CONCEPTS TO JEE ADVANCED (100 QUESTIONS by Grandmaster Bikram Sutradhar)

๐Ÿ“˜ DEFINITE INTEGRATION โ€“ 100 QUESTIONS

By GrandMaster Bikram Sutradhar

F. By-Parts Based Definite Integrals

51. \( \int_{0}^{\pi} x\sin x\,dx \)

52. \( \int_{0}^{\pi/2} x\cos x\,dx \)

53. \( \int_{0}^{\pi} x^2\sin x\,dx \)

54. \( \int_{0}^{\pi/2} x^2\cos x\,dx \)

55. \( \int_{0}^{1} x\ln x\,dx \)

56. \( \int_{0}^{1} x^2\ln x\,dx \)

57. \( \int_{0}^{1} \ln x\,dx \)

58. \( \int_{0}^{1} x^3 e^x\,dx \)

59. \( \int_{0}^{1} x^2 e^{2x}\,dx \)

60. \( \int_{0}^{\pi} x^3\cos x\,dx \)

G. Property-Based & Symmetry (JEE Favourite)

61. \( \int_{0}^{\pi/2} \frac{dx}{1+\sin x} \)

62. \( \int_{0}^{\pi/2} \frac{dx}{1-\cos x} \)

63. \( \int_{0}^{\pi/2} \frac{\sin x}{1+\cos x}\,dx \)

64. \( \int_{0}^{\pi/2} \frac{\cos x}{1-\sin x}\,dx \)

65. \( \int_{0}^{\pi/2} \tan x \sec x\,dx \)

66. \( \int_{0}^{\pi/2} \frac{dx}{\tan x+\sec x} \)

67. \( \int_{0}^{\pi/2} \frac{dx}{\tan x-\sec x} \)

68. \( \int_{0}^{\pi/2} \frac{\sec x}{\tan x}\,dx \)

69. \( \int_{0}^{\pi/2} \frac{\csc x}{\cot x}\,dx \)

70. \( \int_{1}^{e} \sin(\ln x)\,dx \)

H. Logarithmicโ€“Exponential with Definite Limits

71. \( \int_{1}^{e} \cos(\ln x)\,dx \)

72. \( \int_{0}^{1} x\sin(\ln x)\,dx \)

73. \( \int_{0}^{1} x\cos(\ln x)\,dx \)

74. \( \int_{1}^{e} \frac{dx}{x\ln x} \)

75. \( \int_{e}^{e^2} \frac{dx}{x(\ln x)^2} \)

I. Improper & Advanced Rational Definite Integrals

76. \( \int_{0}^{1} \frac{x^3}{x^4+1}\,dx \)

77. \( \int_{1}^{\infty} \frac{dx}{x^4} \)

78. \( \int_{0}^{1} x^2 e^{x^2}\,dx \)

79. \( \int_{0}^{\pi} x^3\sin x\,dx \)

80. \( \int_{0}^{\pi} x^3\cos x\,dx \)

J. Extreme JEE Advanced / Olympiad Level

81. \( \int_{0}^{\infty} \frac{dx}{\sqrt{1+x^4}} \)

82. \( \int_{0}^{1} \sqrt{1+x^4}\,dx \)

83. \( \int_{0}^{1} \frac{dx}{x^2\sqrt{1-x^2}} \)

84. \( \int_{0}^{1} \frac{\sqrt{1-x^2}}{x}\,dx \)

85. \( \int_{1}^{\infty} \frac{dx}{x^2\sqrt{x^2-1}} \)

86. \( \int_{1}^{\infty} \frac{\sqrt{x^2-1}}{x^2}\,dx \)

87. \( \int_{0}^{1} \frac{dx}{\sqrt{x^4+x^2+1}} \)

88. \( \int_{0}^{1} \sqrt{x^4+x^2+1}\,dx \)

89. \( \int_{0}^{1} \frac{x}{\sqrt{x^4+x^2+1}}\,dx \)

90. \( \int_{0}^{1} \frac{dx}{x^4+x^2+1} \)

K. Final High-Concept Definite Integrals

91. \( \int_{0}^{1} \ln(x^2+x+1)\,dx \)

92. \( \int_{0}^{1} x\ln(x^2+1)\,dx \)

93. \( \int_{0}^{\pi/2} \ln(\sin x)\,dx \)

94. \( \int_{0}^{\pi/2} \ln(\cos x)\,dx \)

95. \( \int_{0}^{\pi/2} \ln(\tan x)\,dx \)

96. \( \int_{0}^{a} \frac{dx}{\sqrt{a^2-x^2}} \)

97. \( \int_{0}^{a} \sqrt{a^2-x^2}\,dx \)

98. \( \int_{0}^{a} \frac{x}{\sqrt{a^2-x^2}}\,dx \)

99. \( \int_{0}^{\infty} \frac{dx}{x^2+a^2} \)

100. \( \int_{0}^{1} \frac{x^2}{x^2+a^2}\,dx \)

โœ… COMPLETE โ€” 100 DEFINITE INTEGRALS

  • JEE Main & Advanced
  • Property-based mastery
  • Mock tests & revision books
  • High-level problem-solving

Author: GrandMaster Bikram Sutradhar
Institute: bAstronautWay

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Written By

Full Stack Developer and 5-Time World Record Holder, Grandmaster Bikram Sutradhar

  bAstronautWay :  A Government-Approved Trademark Brand

    SirBikramSutradhar on YouTube

    SirBikramSutradhar is also a Government-Approved Trademark  Brand.                                                               

Written By

Full Stack Developer and 5-Time World Record Holder, Grandmaster Bikram Sutradhar

bAstronautWay

SirBikramSutradhar on YouTube                                                               

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๐Ÿ“š Class 9 Science โ€“ Complete Table of Contents (เคตเคฟเคทเคฏ-เคธเฅ‚เคšเฅ€)

CBSE | NCERT Curriculum | Bilingual (English + Hindi)
By Grandmaster Bikram Sutradhar | SirBikramSutradhar | BASTRONAUTWAY

๐Ÿ”ฌ PHYSICS (เคญเฅŒเคคเคฟเค• เคตเคฟเคœเฅเคžเคพเคจ)

๐Ÿ”ขChapter Title (English)เค…เคงเฅเคฏเคพเคฏ เค•เคพ เคจเคพเคฎ (เคนเคฟเค‚เคฆเฅ€)Page No.
1๏ธโƒฃMotionเค—เคคเคฟ
2๏ธโƒฃForce and Laws of Motionเคฌเคฒ เคคเคฅเคพ เค—เคคเคฟ เค•เฅ‡ เคจเคฟเคฏเคฎ
3๏ธโƒฃGravitationเค—เฅเคฐเฅเคคเฅเคตเคพเค•เคฐเฅเคทเคฃ
4๏ธโƒฃWork and Energyเค•เคพเคฐเฅเคฏ เคคเคฅเคพ เคŠเคฐเฅเคœเคพ
5๏ธโƒฃSoundเคงเฅเคตเคจเคฟ

๐Ÿงช CHEMISTRY (เคฐเคธเคพเคฏเคจ เคตเคฟเคœเฅเคžเคพเคจ)

๐Ÿ”ขChapter Title (English)เค…เคงเฅเคฏเคพเคฏ เค•เคพ เคจเคพเคฎ (เคนเคฟเค‚เคฆเฅ€)Page No.
1๏ธโƒฃMatter in Our Surroundingsเคนเคฎเคพเคฐเฅ‡ เค†เคธ-เคชเคพเคธ เค•เฅ‡ เคชเคฆเคพเคฐเฅเคฅ
2๏ธโƒฃIs Matter Around Us Pure?เค•เฅเคฏเคพ เคนเคฎเคพเคฐเฅ‡ เค†เคธ-เคชเคพเคธ เค•เฅ‡ เคชเคฆเคพเคฐเฅเคฅ เคถเฅเคฆเฅเคง เคนเฅˆเค‚?
3๏ธโƒฃAtoms and Moleculesเคชเคฐเคฎเคพเคฃเฅ เคเคตเค‚ เค…เคฃเฅ
4๏ธโƒฃStructure of the Atomเคชเคฐเคฎเคพเคฃเฅ เค•เฅ€ เคธเค‚เคฐเคšเคจเคพ
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