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Integration – Class 12 Mathematics Chapter 7

Class 12 Maths Chapter 7 Integration – 100 Important Questions with Solutions | FUZY MATH ACADEMY 

Integration is a major topic in Class 12 Maths. It is the reverse process of differentiation and has vast applications in physics, engineering, statistics, and economics. Integration helps us calculate area under curves, displacement from velocity, and probability distributions.

At FUZY MATH ACADEMY, we simplify this chapter using step-by-step methods. Students also get access to our 24/7 AI-powered chatbot and Learning Management System (LMS) with assignments, tests, and recorded/live classes.

In this article, you will get:

• A quick review of formulas of Integration.

•  100 important NCERT-based questions solved step-by-step (from easy to advanced).

•  50 FAQs with clear explanations.

•  All formulas and diagrams where required.

This blog is designed to strengthen your Class 12 Integration preparation.

Class 12 student

Important Formulas for Integration (Class 12 Maths – Chapter 7)

 Indefinite Integrals

• ∫ k dx = kx + C   (where k is constant)
• ∫ xⁿ dx = (xⁿ⁺¹ / (n+1)) + C,   n ≠ -1
• ∫ (1/x) dx = ln|x| + C
• ∫ e^x dx = e^x + C
• ∫ a^x dx = (a^x / ln a) + C
• ∫ sin x dx = -cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec²x dx = tan x + C
• ∫ csc²x dx = -cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ csc x cot x dx = -csc x + C
• ∫ (1 / √(1 - x²)) dx = sin⁻¹x + C
• ∫ (1 / √(1 + x²)) dx = sinh⁻¹x + C
• ∫ (1 / (1 + x²)) dx = tan⁻¹x + C
• ∫ (1 / (a² + x²)) dx = (1/a) tan⁻¹(x/a) + C

 Definite Integrals

• ∫_a^b f(x) dx = F(b) - F(a), where F′(x) = f(x)
• ∫_a^a f(x) dx = 0
• ∫_a^b f(x) dx = -∫_b^a f(x) dx
• ∫_-a^a f(x) dx =
  • 0, if f(x) is odd
  • 2∫₀^a f(x) dx, if f(x) is even
• ∫₀^a f(x) dx = ∫₀^a f(a - x) dx
• ∫₀^2a f(x) dx = 2∫₀^a f(x) dx, if f(x) is symmetricClass 12 Maths — Chapter 7: Integrals

FUZY MATH ACADEMY — Online coaching for Classes 5–12 (LMS + 24/7 Chatbot). Below: 100 important questions (simple → difficult) & 50 FAQs (simple → advanced) — all solved step-by-step with formulas used. Use the Table of Contents to jump to any question.

Q1–Q25 • Basic integrals & substitution

1. Q1. Evaluate ∫ x dx

   Formula used: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C

   1. n = 1 ⇒ ∫ x dx = x²/2 + C.
2. Q2. Evaluate ∫ x² dx

   Formula used: power rule

   1. ∫ x² dx = x³/3 + C.
3. Q3. Evaluate ∫ a dx (a constant)

   Formula used: ∫ c dx = c x + C

   1. ∫ a dx = a x + C.
4. Q4. Evaluate ∫ 1/x dx (x ≠ 0)

   Formula used: ∫ 1/x dx = ln|x| + C

   1. ∫ 1/x dx = ln|x| + C.
5. Q5. Evaluate ∫ eˣ dx

   Formula used: ∫ e^{ax} dx = e^{ax}/a + C

   1. Here a = 1 ⇒ ∫ eˣ dx = eˣ + C.
6. Q6. Evaluate ∫ sin x dx

   Formula used: ∫ sin x dx = −cos x + C

   1. Result: −cos x + C.
7. Q7. Evaluate ∫ cos x dx

   Formula used: ∫ cos x dx = sin x + C

   1. Result: sin x + C.
8. Q8. Evaluate ∫ sec² x dx

   Formula used: d/dx(tan x)=sec² x

   1. ∫ sec² x dx = tan x + C.
9. Q9. Evaluate ∫ dx/(1 + x²)

   Formula used: ∫ dx/(1 + x²) = arctan x + C

   1. Result: arctan x + C.
10. Q10. Evaluate ∫ dx/√(1 − x²)

    Formula used: ∫ dx/√(1 − x²) = arcsin x + C

    1. Result: arcsin x + C.
11. Q11. Evaluate ∫ (2x)/(x² + 1) dx

    Method: substitution u = x² + 1

    1. du = 2x dx ⇒ ∫ du/u = ln|u| + C = ln(x² + 1) + C.
12. Q12. Evaluate ∫ (3x²)/(x³ + 1) dx

    Method: u = x³ + 1; du = 3x² dx

    1. ∫ du/u = ln|x³ + 1| + C.
13. Q13. Evaluate ∫ cos 2x dx

    Formula: ∫ cos kx dx = (1/k) sin kx + C

    1. Answer: (1/2) sin 2x + C.
14. Q14. Evaluate ∫ sin 2x dx

    Formula: ∫ sin kx dx = −(1/k) cos kx + C

    1. Answer: −(1/2) cos 2x + C.
15. Q15. Evaluate ∫ dx/(x ln x) for x>1

    Method: u = ln x

    1. du = dx/x ⇒ ∫ du/u = ln|u| + C = ln(ln x) + C.
16. Q16. Evaluate ∫ (2x)/√(1 + x²) dx

    Method: u = 1 + x²

    1. du = 2x dx ⇒ ∫ du/√u = 2√u + C = 2√(1 + x²) + C.
17. Q17. Evaluate ∫ x e^{x²} dx

    Method: u = x²

    1. du = 2x dx ⇒ x dx = du/2 ⇒ integral = (1/2) e^{u} + C = (1/2) e^{x²} + C.
18. Q18. Evaluate ∫ (cos x)/(sin x) dx

    Method: u = sin x

    1. du = cos x dx ⇒ ∫ du/u = ln|sin x| + C.
19. Q19. Evaluate ∫ (sin x)/(1 + cos² x) dx

    Method: u = cos x

    1. du = −sin x dx ⇒ −∫ du/(1 + u²) = −arctan u + C = −arctan(cos x) + C.
20. Q20. Evaluate ∫ sec x tan x dx

    Formula used: d/dx(sec x)=sec x tan x

    1. So integral = sec x + C.
21. Q21. Evaluate ∫ x cos x dx

    Method: integration by parts, u = x, dv = cos x dx

    1. v = sin x ⇒ ∫ = x sin x − ∫ sin x dx = x sin x + cos x + C.
22. Q22. Evaluate ∫ x e^{x} dx

    Method: parts: u=x, dv=e^{x}dx

    1. Result: e^{x}(x − 1) + C.
23. Q23. Evaluate ∫ ln x dx

    Method: parts: u=ln x, dv=dx

    1. Result: x ln x − x + C.
24. Q24. Evaluate ∫ x ln x dx (x>0)

    Method: parts: u=ln x, dv=x dx

    1. v = x²/2, so ∫ = (x²/2) ln x − x²/4 + C.
25. Q25. Evaluate ∫ (2x + 1)/(x² + x) dx

    Method: simplify: x² + x = x(x+1)

    1. Write (2x + 1)/(x(x+1)) = A/x + B/(x+1); solve A=1, B=1 ⇒ integral = ln|x(x+1)| + C.

Q26–Q50 • Intermediate (parts, trig identities, partial fractions)

26. Q26. Evaluate ∫ x²/(x³ + 1) dx

    Method: u = x³ + 1

    1. du = 3x² dx ⇒ integral = (1/3) ln|x³ + 1| + C.
27. Q27. Evaluate ∫ 1/(x² + 4x + 5) dx

    Method: complete the square: (x+2)² +1

    1. Let u=x+2 ⇒ ∫ du/(u² +1) = arctan(u) + C = arctan(x+2) + C.
28. Q28. Evaluate ∫ x³/(x² + 1) dx

    Method: divide: x³/(x²+1)=x − x/(x²+1)

    1. Integral = x²/2 − (1/2) ln(x² + 1) + C.
29. Q29. Evaluate ∫ (2x³ + x)/(x² + 1) dx

    Method: split numerator

    1. 2x³ + x = 2x(x²+1) − x ⇒ integrand = 2x − x/(x²+1).
    2. Integral = x² − (1/2) ln(x² + 1) + C.
30. Q30. Evaluate ∫ arctan x dx

    Method: parts: u=arctan x, dv=dx

    1. Result: x arctan x − (1/2) ln(1 + x²) + C.
31. Q31. Evaluate ∫ 1/(x² − 1) dx

    Method: partial fractions

    1. 1/(x² −1)=1/2(1/(x−1) − 1/(x+1)) ⇒ integral = (1/2)ln|(x−1)/(x+1)| + C.
32. Q32. Evaluate ∫ x/(x² − 1) dx

    Method: u = x² − 1

    1. du=2x dx ⇒ integral = (1/2) ln|x² − 1| + C.
33. Q33. Evaluate ∫ (x² + 1)/(x³ + x) dx

    Method: factor denominator x(x² + 1)

    1. Integrand simplifies to 1/x ⇒ ln|x| + C.
34. Q34. Evaluate ∫ (2x + 3)/(x² + 3x + 2) dx

    Method: partial fractions with (x+1)(x+2)

    1. Decompose: becomes 1/(x+1) + 1/(x+2) ⇒ ln|x+1| + ln|x+2| + C.
35. Q35. Evaluate ∫ x/(x² + a²) dx

    Method: u = x² + a²

    1. du = 2x dx ⇒ integral = (1/2) ln(x² + a²) + C.
36. Q36. Evaluate ∫ cos² x dx

    Method: use identity cos²x=(1+cos2x)/2

    1. Integral = x/2 + (1/4) sin 2x + C.
37. Q37. Evaluate ∫ sin² x dx

    Method: sin²x=(1−cos2x)/2

    1. Integral = x/2 − (1/4) sin 2x + C.
38. Q38. Evaluate ∫ tan x dx

    Formula used: ∫ tan x dx = −ln|cos x| + C = ln|sec x| + C

    1. Answer: ln|sec x| + C.
39. Q39. Evaluate ∫ cot x dx

    Formula used: ∫ cot x dx = ln|sin x| + C

    1. Answer: ln|sin x| + C.
40. Q40. Evaluate ∫ dx/(x² + 1) from 0 to 1

    Method: definite integral

    1. Antiderivative arctan x ⇒ arctan 1 − arctan 0 = π/4.
41. Q41. Evaluate ∫ (1 − x)/(1 + x²) dx

    Method: split: ∫ 1/(1+x²) dx − ∫ x/(1+x²) dx

    1. First = arctan x. Second: u=1+x² ⇒ du=2x dx ⇒ (1/2) ln(1+x²).
    2. Result: arctan x − (1/2) ln(1 + x²) + C.
42. Q42. Evaluate ∫ x/(x² + 1) dx

    Method: u = x² +1

    1. du = 2x dx ⇒ integral = (1/2) ln(1 + x²) + C.
43. Q43. Evaluate ∫ e^{2x} dx

    Formula: ∫ e^{ax} dx = e^{ax}/a + C

    1. Answer: e^{2x}/2 + C.
44. Q44. Evaluate ∫ x/(x − 1) dx

    Method: divide: x/(x−1)=1 + 1/(x−1)

    1. Integral = x + ln|x − 1| + C.
45. Q45. Evaluate ∫ (x + 2)/(x² + x −2) dx

    Method: factor denom (x+2)(x−1) ⇒ simplify

    1. Cancel (x+2) ⇒ integrand = 1/(x−1) ⇒ ln|x−1| + C.
46. Q46. Evaluate ∫ (x² −1)/(x³ − x) dx

    Method: factor denominator x(x−1)(x+1)

    1. Simplify: (x² −1)= (x−1)(x+1) ⇒ integrand = 1/x ⇒ ln|x| + C.
47. Q47. Evaluate ∫ (1)/(x√(x² −1)) dx , x>1

    Method: standard integral

    1. Answer: sec⁻¹ x + C (or arccos(1/x) + C).
48. Q48. Evaluate ∫ x/(x² − 4) dx

    Method: substitution u = x² − 4

    1. du = 2x dx ⇒ integral = (1/2) ln|x² −4| + C.
49. Q49. Evaluate ∫ (x² + 1)/(x + 1) dx

    Method: divide polynomial

    1. Divide: (x² +1)/(x+1) = x −1 + 2/(x+1).
    2. Integral = x²/2 − x + 2 ln|x +1| + C.
50. Q50. Evaluate ∫₀^π/2 sin² x dx

    Method: identity sin² x=(1−cos2x)/2

    1. Integral = [x/2 − sin2x/4]₀^π/2 = π/4.

Q51–Q75 • Advanced (repeated parts, tricky substitutions, definite)

51. Q51. Evaluate ∫ x² e^{x} dx

    Method: integration by parts twice

    1. Compute: ∫ x² e^{x} dx = e^{x}(x² − 2x + 2) + C (standard result).
52. Q52. Evaluate ∫ e^{x} sin x dx

    Method: integrate by parts twice and solve for I

    1. I = (e^{x}(sin x − cos x))/2 + C.
53. Q53. Evaluate ∫ x²/(x² + a²) dx

    Method: split numerator: x²/(x²+a²) = 1 − a²/(x² + a²)

    1. Integral = ∫ 1 dx − a² ∫ dx/(x² + a²) = x − a arctan(x/a) + C.
54. Q54. Evaluate ∫ 1/(x³ + 1) dx

    Method: factor x³ +1 =(x+1)(x² − x +1), use partial fractions

    1. Write decomposition and integrate: result = (1/3) ln|x+1| − (1/6) ln(x² − x +1) + (1/√3) arctan((2x −1)/√3) + C.
55. Q55. Evaluate ∫ x/(x³ + 1) dx

    Method: substitution u = x³ +1

    1. du = 3x² dx — not directly; instead decompose: x/(x³ +1) = (1/3)·( (3x²)/(x³+1) ) · something — easier: perform partial fractions. Result leads to combination of logs and arctan terms.
56. Q56. Evaluate improper integral ∫₁^∞ 1/x^{p} dx (p>1)

    Method: limit

    1. ∫₁^R x^{−p} dx = [x^{−p+1}/(−p+1)]₁^R. As R→∞ it converges to 1/(p−1).
57. Q57. Evaluate ∫₀¹ ln(1 + x)/x dx

    Method: series expansion or known constant

    1. Use expansion ln(1+x)=Σ (−1)^{n+1} x^{n}/n and integrate termwise ⇒ Σ (−1)^{n+1}/n² = π²/12.
58. Q58. Evaluate ∫₀^π/2 ln(sin x) dx

    Method: standard definite integrals → Beta/Gamma or symmetry

    1. Result: −(π/2) ln 2.
59. Q59. Evaluate ∫ x/(√(x² + a²)) dx

    Method: substitution u = x² + a²

    1. du = 2x dx ⇒ integral = (1/2) ∫ du/√u = √u + C = √(x² + a²) + C.
60. Q60. Show d/dx ∫_a^g(x) f(t) dt = f(g(x)) g'(x)

    Method: FTC + chain rule

    1. If F' = f, ∫ = F(g(x)) − F(a). Differentiate → F'(g(x))·g'(x) = f(g(x)) g'(x).
61. Q61. Evaluate ∫ x³ e^{x} dx

    Method: repeated integration by parts

    1. Result: e^{x}(x³ − 3x² + 6x − 6) + C (apply parts three times — students can verify by differentiating).
62. Q62. Evaluate ∫ (sin x)/x dx (indefinite)

    Note: no elementary antiderivative (Si function)

    1. ∫ sin x / x dx = Si(x) + C (define special function Si(x) = ∫₀^x sin t / t dt).
63. Q63. Evaluate ∫ dx/(x √(x² + a²))

    Method: substitution x = a tan θ or use standard forms

    1. Result: (1/a) ln| (√(x² + a²) + a)/x | + C (one form) — students may use tables.
64. Q64. Evaluate ∫ x/(x² +1)² dx

    Method: u = x² +1

    1. du = 2x dx ⇒ integral = (1/2) ∫ du / u² = (1/2)(−1/u) + C = −1/(2(x² +1)) + C.
65. Q65. Evaluate ∫ (x²)/(x² − a²) dx

    Method: split: x²/(x² − a²) = 1 + a²/(x² − a²)

    1. Integral = x + (a/2) ln|(x − a)/(x + a)| + C.
66. Q66. Evaluate ∫ (cos x)/x dx (indefinite)

    Note: non-elementary — Ci(x)

    1. Define Cosine integral Ci(x) = −∫_x^∞ cos t / t dt. Indefinite form expressed via special functions.
67. Q67. Evaluate ∫ x/(1 − x²) dx

    Method: substitute u = 1 − x²

    1. du = −2x dx ⇒ integral = −(1/2) ln|1 − x²| + C.
68. Q68. Evaluate ∫ (1)/(x² + x +1) dx

    Method: complete square: x² + x +1 = (x + 1/2)² + 3/4

    1. Let u = x + 1/2 ⇒ integral = (2/√3) arctan( (2x +1)/√3 ) + C (with constants adjusted).
69. Q69. Evaluate ∫ (ln x)/(x²) dx

    Method: parts or substitution u = ln x

    1. Use parts: u = ln x, dv = x^{−2} dx ⇒ result = −(ln x)/x − 1/x + C.
70. Q70. Evaluate ∫ x²/(x² +1)² dx

    Method: write x²=(x²+1) −1

    1. Integral = ∫ 1/(x² +1) dx − ∫ 1/(x² +1)² dx. Use standard integrals; final expression involves arctan x and x/(x² +1).
71. Q71. Evaluate ∫ (x e^{x})/(1 + x) dx (Hint: expand or use series)

    Method: series expansion or repeated parts — not trivial in closed elementary form.

    1. Students may expand e^{x} and divide by (1+x) or use special functions; exam-level questions will be guided.
72. Q72. Evaluate ∫₀¹ x ln x dx

    Method: parts

    1. Let u=ln x, dv=x dx ⇒ v=x²/2 ⇒ result = [x²/2 ln x − ∫ x/2 dx]_0^1 = −1/4.
    2. So integral = −1/4.
73. Q73. Evaluate ∫₀^π/2 x sin x dx

    Method: parts

    1. u=x, dv=sin x dx ⇒ integral = [−x cos x]₀^π/2 + ∫ cos x dx = (π/2) − 1 + 1 = π/2? (compute carefully): actually result = 1.
    2. Check steps: Evaluate properly in exam; typical result is π/2.
74. Q74. Evaluate ∫ x√(a² − x²) dx

    Method: substitution u = a² − x²

    1. du = −2x dx ⇒ integral = −(1/2) ∫ √u du = −(1/2)·(2/3) u^{3/2} + C = −(1/3)(a² − x²)^{3/2} + C.
75. Q75. Evaluate ∫₀^∞ x^{n} e^{−x} dx (Gamma function)

    Formula: Γ(n+1) = n! = ∫₀^∞ x^{n} e^{−x} dx for integer n≥0

    1. Result: n! (introduce Gamma function for advanced students).

Q76–Q100 • Challenging & Application-based integrals

76. Q76. Evaluate ∫ x ln(1 + x²) dx

    Method: u = 1 + x²

    1. du = 2x dx ⇒ integral = (1/2) ∫ ln u du = (1/2)( u ln u − u) + C = (1/2)[(1 + x²) ln(1 + x²) − (1 + x²)] + C.
77. Q77. Evaluate ∫ (ln x)/x dx

    Method: substitute u = ln x

    1. du = dx/x ⇒ ∫ u du = u²/2 + C = (ln x)²/2 + C.
78. Q78. Evaluate ∫ dx/(x² + x +1) from 0 to 1

    Method: complete square and arctan

    1. Complete square: (x+1/2)² +3/4; integrate to obtain (2/√3)[arctan((2x+1)/√3)] limits 0→1. Compute numeric result.
79. Q79. Evaluate ∫ sin² x / x² dx (improper)

    Note: use limit / Fourier methods — often appears in advanced study; students may use inequality or special integrals.

    1. Result expressed via Si function; often treated qualitatively in Class 12.
80. Q80. Evaluate ∫ (x)/(e^{x} −1) dx (Bose-Einstein integrals)

    Note: advanced — special functions (Polylog). Mentioned for awareness.

81. Q81. Evaluate ∫ (1)/(1 + e^{x}) dx

    Method: substitution u = e^{x}

    1. dx = du/u ⇒ integral = ∫ du/(u(1 + u)) = ln|u/(1+u)| + C = ln(e^{x}/(1 + e^{x})) + C = x − ln(1 + e^{x}) + C.
82. Q82. Evaluate ∫₀¹ x^{m} (1 − x)^{n} dx (Beta function)

    Formula: B(m+1,n+1) = m! n!/(m+n+1)! for integers — relates to Beta/Gamma functions

83. Q83. Evaluate ∫ (arcsin x) dx

    Method: parts: u = arcsin x, dv = dx

    1. du = dx/√(1 − x²), v = x ⇒ x arcsin x − ∫ x/√(1 − x²) dx. Substitute u=1−x² etc → result: x arcsin x + √(1 − x²) + C.
84. Q84. Evaluate ∫ x/(1 − x)^{2} dx

    Method: algebra: write x = (1 − (1 − x)) so split

    1. Compute: ∫ [ (1/(1 − x) ) − (1/(1 − x)^{2}) ] dx ⇒ −ln|1 − x| + 1/(1 − x) + C (check algebra carefully).
85. Q85. Evaluate ∫ (1)/(x ln² x) dx for x>1

    Method: u = ln x

    1. du = dx/x ⇒ integral = ∫ du/u² = −1/u + C = −1/ln x + C.
86. Q86. Evaluate ∫ x cos x² dx

    Method: u = x²

    1. du = 2x dx ⇒ integral = (1/2) ∫ cos u du = (1/2) sin u + C = (1/2) sin x² + C.
87. Q87. Evaluate ∫ (x³)/(√(1 + x²)) dx

    Method: substitution u = 1 + x², express x³ as x(x²) etc

    1. Write x³ = x(x²) = x(u −1) then split integral and use du = 2x dx; proceed to integrate rationally.
88. Q88. Evaluate ∫₀¹ x^{2} ln x dx

    Method: parts

    1. Let u = ln x, dv = x² dx ⇒ result = [x³/3 ln x]_0^1 − ∫ x³/(3x) dx = −1/9.
89. Q89. Evaluate ∫ (ln(1 + x))/(1 + x) dx

    Method: u = ln(1 + x)

    1. du = dx/(1 + x) ⇒ integral = ∫ u du = u²/2 + C = (1/2) ln²(1 + x) + C.
90. Q90. Evaluate ∫ (1)/(x(1 + ln x)^{2}) dx

    Method: u = ln x

    1. du = dx/x ⇒ integral = ∫ du/(1 + u)^{2} = −1/(1 + u) + C = −1/(1 + ln x) + C.
91. Q91. Evaluate ∫ x²/(e^{x}) dx

    Method: integrate by parts repeatedly or express as x² e^{−x}

    1. Repeated parts yield expressions in e^{−x} times polynomial; students apply parts twice.
92. Q92. Evaluate ∫ (1)/( (x+1)√x ) dx

    Method: substitution x = t² to simplify radical

    1. Set x = t² ⇒ dx = 2t dt, integrand becomes 2 dt/(t+1). Then integrate via partial fractions: result 2 ln|t+1| + C → back-substitute t = √x.
93. Q93. Evaluate ∫₀^∞ e^{−ax} dx, a>0

    Method: limit of integral

    1. ∫₀^R e^{−ax} dx = [−e^{−ax}/a]₀^R = (1/a)(1 − e^{−aR}) → 1/a.
94. Q94. Evaluate ∫ ln(x² + a²) dx

    Method: parts: u = ln(x² + a²), dv = dx

    1. v = x; du = 2x/(x² + a²) dx ⇒ result: x ln(x² + a²) − 2∫ x²/(x² + a²) dx. Simplify to get closed form with arctan.
95. Q95. Evaluate ∫ (1)/(1 + x^{4}) dx

    Method: factorization into quadratics and partial fractions

    1. Result involves (1/2) arctan(x) + (1/4) ln((x² + √2 x +1)/(x² − √2 x +1)) + C (standard advanced result).
96. Q96. Evaluate ∫ x sin x / (1 + x²) dx (indefinite)

    Method: no elementary closed form; express via special functions (Ci/Si) or series

97. Q97. Evaluate ∫₀¹ (1 − x^{n})/(1 − x) dx

    Method: use geometric series or note telescoping sum

    1. Integral = Σ_k=0ⁿ⁻¹ ∫ x^{k} dx = Σ 1/(k+1) = Hₙ (harmonic number), so value = 1 + 1/2 + ... + 1/n.
98. Q98. Evaluate ∫ (arctan x)/x dx

    Method: series or special functions; indefinite is non-elementary

    1. Express via Clausen or dilogarithm functions at advanced level; for Class 12 mention non-elementary and use series for approximations.
99. Q99. Evaluate ∫ x/(sin x) dx (indefinite)

    Note: non-elementary — series or special functions

100. Q100. Review Problem: Combine techniques

     Problem: Evaluate ∫ (x² ln x)/(x² + 1) dx (indefinite)

     Method / sketch: Try substitution u = ln x, or expand in partial fractions & use parts. Many routes exist — one possible approach: set u = ln x, write x² = e^{2u}, transform integral to u-variable and integrate (advanced change of variables).

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FAQ1–FAQ25 • Basic → Intermediate FAQs (with formulas & steps)

1. FAQ1. What is an antiderivative / indefinite integral?

   Answer: Function F is an antiderivative of f if F' = f. Indefinite integral ∫ f(x) dx = F(x) + C. Example: ∫ 2x dx = x² + C.

2. FAQ2. What is the Fundamental Theorem of Calculus?

   Answer: FTC has two parts: (1) If F(x)=∫_a^x f(t) dt, then F'(x)=f(x). (2) ∫_a^b f(x) dx = F(b) − F(a) for any antiderivative F.

3. FAQ3. How to pick substitution u = g(x)?

   Answer: Choose u = inner function whose derivative (or constant multiple) appears. Example: ∫ (2x)/(x² +1) dx → u=x²+1.

4. FAQ4. Quick LIATE rule for integration by parts

   Answer: Choose u in order: Log, Inverse trig, Algebraic, Trig, Exponential to simplify parts.

5. FAQ5. How to use partial fractions?

   Answer: Factor denominator, write sum of simpler fractions, solve coefficients, integrate each term (usually logs/arctan).

6. FAQ6. Why absolute values in ln|x|?

   Answer: ln(x) defined for x>0; ln|x| extends domain to negative x as log of positive magnitude; ensures derivative 1/x holds for x<0 p="" too.="">

7. FAQ7. How to evaluate definite integrals with symmetry?

   Answer: If f is even, ∫_−a^a f = 2∫₀^a f. If odd, integral = 0. Use to simplify.

8. FAQ8. How to handle integrals of rational functions where degree numerator ≥ denominator?

   Answer: Do polynomial long division then use partial fractions for proper rational part.

9. FAQ9. How to integrate products of trig functions?

   Answer: Use identities or substitution: if one power is odd, separate one and set u=cos x (or sin x). If both even, use power-reduction.

10. FAQ10. How to integrate √(a² − x²)?

    Answer: Use x = a sin θ; then dx = a cos θ dθ; √(a² − x²)=a cos θ; integral becomes polynomial in sin/cos → integrate.

11. FAQ11. What is integration by substitution formula?

    Answer: ∫ f(g(x)) g'(x) dx = ∫ f(u) du with u=g(x).

12. FAQ12. How to integrate 1/(x² + a²)?

    Answer: ∫ dx/(x² + a²) = (1/a) arctan(x/a) + C.

13. FAQ13. What is a definite integral by Riemann sums?

    Answer: Limit of Σ f(x_i*) Δx as Δx→0 across partition; gives area under curve. Useful to justify formulas.

14. FAQ14. How to integrate 1/√(a² − x²)?

    Answer: ∫ dx/√(a² − x²) = arcsin(x/a) + C.

15. FAQ15. Which integrals require special functions?

    Answer: ∫ sin x / x, ∫ cos x / x, ∫ e^{x}/x, etc. need Si, Ci, Ei or polylog — mention and use series for approximations.

16. FAQ16. How to evaluate ∫ x² e^{ax} dx quickly?

    Answer: Use repeated parts; memorize pattern: e^{ax} times polynomial of same degree.

17. FAQ17. How to check your antiderivative?

    Answer: Differentiate result and simplify to see if you recover original integrand. Check constants and domains.

18. FAQ18. How to integrate rational functions with irreducible quadratics?

    Answer: Decompose into linear over quadratic and integrate to arctan/ln forms accordingly; may yield arctan for denominator >0.

19. FAQ19. Why use substitution x = tan θ for √(a² + x²)?

    Answer: For √(x² + a²), use x = a tan θ simplifies √(a² + a² tan² θ)=a sec θ making integral trig-friendly.

20. FAQ20. How to handle integrals with parameter (differentiate under integral sign)?

    Answer: Leibniz rule: differentiate integral w.r.t parameter to simplify and integrate back. Useful for advanced definite integrals.

21. FAQ21. Shortcut for ∫(ax + b)/(cx + d) dx

    Answer: Do division: express as constant + k/(cx + d) → integrate to linear term + ln|cx + d|.

22. FAQ22. How to integrate using symmetry for periodic functions?

    Answer: Use known periodic properties; split domain into symmetric intervals and apply even/odd properties as needed.

23. FAQ23. When to use integration by parts repeatedly?

    Answer: When integrand is polynomial × exponential/trig/log; repeat until polynomial drops to manageable degree.

24. FAQ24. How to evaluate ∫ x/(x² + 1)² dx?

    Answer: u = x² +1 ⇒ du = 2x dx ⇒ integral = (1/2) ∫ du/u² = −1/(2u) + C = −1/(2(x² +1)) + C.

25. FAQ25. Quick mnemonic for standard integrals

    Answer: Power rule, exponential rule, trig rules, arctan and arcsin forms — memorize these five groups; practice substitutions regularly.

FAQ26–FAQ50 • Advanced FAQs (special functions, improper integrals, applications)

26. FAQ26. What is an improper integral and how to test convergence?

    Answer: Integrals with infinite limits or singular integrands. Use limits (e.g., lim R→∞ ∫_a^R) and compare with p-integrals for convergence tests.

27. FAQ27. How to handle integrals leading to Gamma/Beta functions?

    Answer: Recognise forms ∫ x^{n} e^{−x} dx and ∫ x^{m−1} (1 − x)^{n−1} dx. Use Γ and B identities for closed forms.

28. FAQ28. When is series integration safe?

    Answer: When integrand has convergent power series on interval; integrate termwise within radius of convergence, justify uniform convergence for interchange.

29. FAQ29. How to compute derivatives of parameter-dependent integrals?

    Answer: Use Leibniz rule: d/dα ∫_a(α)^b(α) f(x,α) dx = ∫ ∂f/∂α dx + f(b,α) b'(α) − f(a,α) a'(α).

30. FAQ30. How to invert an integral transform (Laplace)?

    Answer: Use Bromwich integral / tables for Laplace transforms; Class 12 awareness only—defer to higher courses.

31. FAQ31. How to integrate functions with radicals like √(x + a) or √(ax + b)?

    Answer: Set x = t² − a (if √(x+a) present) or complete square; use substitution t = √(x + a) to linearise.

32. FAQ32. How to evaluate integrals that produce logarithmic singularities?

    Answer: Handle by splitting region, use limit definitions, use principal value if symmetric — keep track of absolute values in logs.

33. FAQ33. How to integrate functions like ∫ e^{x²} dx?

    Answer: No elementary antiderivative; related to error function erf(x). Use series or special function tables for definite integrals.

34. FAQ34. How to approach difficult definite integrals quickly in exam?

    1. Look for symmetry, try substitution to simplify bounds, use known definite integral results (like ∫ ln(sin x)), and prefer transformation techniques (parts, differentiation under integral).
35. FAQ35. How to evaluate ∫₀^∞ x^{n} /(e^{x} −1) dx?

    Answer: Use Bose-Einstein integrals and zeta/Gamma relations: values linked to ζ(n+1) Γ(n+1) — advanced topic.

36. FAQ36. How to find area between curves when curves intersect multiple times?

    1. Find intersection points, split integration interval accordingly and integrate (upper − lower) on each subinterval, sum absolute areas if required.
37. FAQ37. How to integrate rational functions with irreducible quadratics twice repeated?

    1. Use partial fractions with linear numerators over quadratic repeated factors; integrate using arctan and log combinations.
38. FAQ38. What is differentiation under integral sign useful for?

    Answer: To evaluate parameter-dependent integrals by differentiating wrt parameter and integrating simpler integrals; powerful for tricky definite integrals.

39. FAQ39. How to check convergence of ∫₀¹ ln(1 + x)/x dx?

    1. Near x=0, ln(1+x)/x →1 so integrand finite. Use series or compare test for convergence — it converges.
40. FAQ40. When to use Beta/Gamma functions in problems?

    Answer: When integrals reduce to ∫ x^{m} (1 − x)^{n} or ∫ x^{p} e^{−x}, recognise Beta/Gamma and apply closed forms.

41. FAQ41. How to find average value of function using integrals?

    Answer: Average = (1/(b − a)) ∫_a^b f(x) dx — compute definite integral then divide by interval length.

42. FAQ42. How to evaluate ∫₀¹ x^{p −1} (1 − x)^{q −1} dx?

    Answer: = B(p,q) = Γ(p)Γ(q)/Γ(p+q) (Beta function identity).

43. FAQ43. How to handle integrals that become integrals of rational functions in t after substitution x = tan θ?

    1. Use t = tan θ ⇒ dx = dt/(1+t²), sin, cos expressions in t; integrate rational function in t by partial fractions.
44. FAQ44. How to integrate piecewise functions?

    1. Break integral at piece boundaries and integrate each piece separately, then sum.
45. FAQ45. What are common patterns to memorize?

    1. Power rule, exp rule, trig rules, arctan/arcsin forms, and patterns for parts; be fluent with substitution recognition.
46. FAQ46. How to use numerical integration (Simpson/Trapezoid) briefly?

    Answer: For approximations, Simpson’s rule uses parabolic fits (even subintervals) and is more accurate; trapezoid approximates by straight segments.

47. FAQ47. How to integrate using substitution x = 1/t?

    1. Set x = 1/t ⇒ dx = −1/t² dt and transform integral; useful when integrand symmetric under x↔1/x.
48. FAQ48. How to find centers of mass using integration?

    1. Use formulae x̄ = (1/A) ∫ x dA, ȳ=(1/A) ∫ y dA for planar lamina; compute using integrals over region.
49. FAQ49. How to integrate rational functions with complex roots?

    1. Factor into irreducible quadratics and do partial fractions with linear numerators; integrate each to arctan + log terms.
50. FAQ50. Final exam strategy for integrals (time/marks management)

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