JEE (Main) Mathematics - Integral Calculus MCQs

46. The integral of \( \sec^2(x) \) is:
    • a) \( \tan(x) + C \)
    • b) \( \cos(x) + C \)
    • c) \( \sec(x) + C \)
    • d) \( -\cos(x) + C \)
    Answer: a) \( \tan(x) + C \)
47. The integral \( \int \frac{1}{x} \, dx \) gives:
    • a) \( \ln(x) + C \)
    • b) \( \frac{1}{x} + C \)
    • c) \( x + C \)
    • d) \( \ln(1+x) + C \)
    Answer: a) \( \ln(x) + C \)
48. The integral of \( \cot(x) \) is:
    • a) \( \ln(\sin(x)) + C \)
    • b) \( -\ln(\sin(x)) + C \)
    • c) \( \ln(\cos(x)) + C \)
    • d) \( -\ln(\cos(x)) + C \)
    Answer: b) \( -\ln(\sin(x)) + C \)
49. Which of the following integrals is evaluated using integration by parts?
    • a) \( \int e^x \, dx \)
    • b) \( \int x^2 \, dx \)
    • c) \( \int x \ln(x) \, dx \)
    • d) \( \int \frac{1}{x^2 + 1} \, dx \)
    Answer: c) \( \int x \ln(x) \, dx \)
50. The area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 2 \) is:
    • a) 1
    • b) 2
    • c) 3
    • d) 4
    Answer: b) 2
51. The integral \( \int \sin^2(x) \, dx \) can be evaluated using:
    • a) Integration by parts
    • b) Trigonometric identities
    • c) Partial fractions
    • d) Substitution
    Answer: b) Trigonometric identities
52. The integral of \( \frac{1}{x^2 + 4} \) is:
    • a) \( \frac{1}{2} \ln(x^2 + 4) + C \)
    • b) \( \frac{1}{2} \tan^{-1}\left( \frac{x}{2} \right) + C \)
    • c) \( \ln(x^2 + 4) + C \)
    • d) \( \tan^{-1}(x) + C \)
    Answer: b) \( \frac{1}{2} \tan^{-1}\left( \frac{x}{2} \right) + C \)
53. The integral of \( \sec^2(x) \) is:
    • a) \( \tan(x) + C \)
    • b) \( \sec(x) + C \)
    • c) \( \sin(x) + C \)
    • d) \( \cos(x) + C \)
    Answer: a) \( \tan(x) + C \)
54. The integral of \( \frac{1}{x(x+1)} \) is solved using:
    • a) Substitution
    • b) Integration by parts
    • c) Partial fractions
    • d) Trigonometric identities
    Answer: c) Partial fractions
55. The integral of \( \sin(x) \cos(x) \) is:
    • a) \( \frac{1}{2} \sin^2(x) + C \)
    • b) \( \frac{1}{2} \cos^2(x) + C \)
    • c) \( -\cos(x) + C \)
    • d) \( \sin^2(x) + C \)
    Answer: a) \( \frac{1}{2} \sin^2(x) + C \)
56. Which of the following is true for definite integrals?
    • a) The limits of integration must always be positive.
    • b) The integral always gives a positive value.
    • c) The value of a definite integral is independent of the limits.
    • d) The definite integral represents the area under the curve.
    Answer: d) The definite integral represents the area under the curve.
57. The integral of \( \ln(x) \) is:
    • a) \( x \ln(x) - x + C \)
    • b) \( \ln(x) + C \)
    • c) \( x \ln(x) + C \)
    • d) \( \frac{1}{x} + C \)
    Answer: a) \( x \ln(x) - x + C \)
58. The area between the curve \( y = \cos(x) \) and the x-axis from \( x = 0 \) to \( x = \pi \) is:
    • a) 0
    • b) 1
    • c) 2
    • d) 4
    Answer: b) 2
59. The integral of \( \frac{1}{\sqrt{1-x^2}} \) is:
    • a) \( \sin^{-1}(x) + C \)
    • b) \( \cos^{-1}(x) + C \)
    • c) \( \tan^{-1}(x) + C \)
    • d) \( \ln(1+x^2) + C \)
    Answer: a) \( \sin^{-1}(x) + C \)
60. The integral of \( \sec(x) \tan(x) \) is:
    • a) \( \ln(\sec(x)) + C \)
    • b) \( \cos(x) + C \)
    • c) \( \tan(x) + C \)
    • d) \( \sin(x) + C \)
    Answer: a) \( \ln(\sec(x)) + C \)

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