JEE (Main) Mathematics - Integral Calculus MCQs

61. The integral of \( \frac{1}{x} \) with respect to \( x \) is:
    • a) \( \ln(x) + C \)
    • b) \( \ln(|x|) + C \)
    • c) \( \frac{1}{x^2} + C \)
    • d) \( x \ln(x) + C \)
    Answer: b) \( \ln(|x|) + C \)
62. The integral of \( \sin(x) \) with respect to \( x \) is:
    • a) \( -\cos(x) + C \)
    • b) \( \cos(x) + C \)
    • c) \( \sin(x) + C \)
    • d) \( -\sin(x) + C \)
    Answer: a) \( -\cos(x) + C \)
63. Which of the following integrals requires substitution to solve?
    • a) \( \int x^2 e^x \, dx \)
    • b) \( \int \ln(x) \, dx \)
    • c) \( \int \frac{1}{x^2 + 1} \, dx \)
    • d) \( \int \sin(x) \, dx \)
    Answer: a) \( \int x^2 e^x \, dx \)
64. The integral \( \int e^x \, dx \) is:
    • a) \( e^x + C \)
    • b) \( e^{2x} + C \)
    • c) \( x e^x + C \)
    • d) \( \ln(x) + C \)
    Answer: a) \( e^x + C \)
65. The integral \( \int \sec^2(x) \, dx \) is:
    • a) \( \ln(\tan(x)) + C \)
    • b) \( \tan(x) + C \)
    • c) \( \cos(x) + C \)
    • d) \( \ln(\sec(x)) + C \)
    Answer: b) \( \tan(x) + C \)
66. Which of the following is the result of the integral \( \int \frac{1}{1 + x^2} \, dx \)?
    • a) \( \arctan(x) + C \)
    • b) \( \ln(1 + x^2) + C \)
    • c) \( \arcsin(x) + C \)
    • d) \( \frac{x}{1 + x^2} + C \)
    Answer: a) \( \arctan(x) + C \)
67. The integral \( \int \cos^2(x) \) can be simplified using:
    • a) Substitution
    • b) Integration by parts
    • c) Trigonometric identities
    • d) Partial fractions
    Answer: c) Trigonometric identities
68. The integral \( \int x \, e^x \, dx \) can be solved using:
    • a) Substitution
    • b) Integration by parts
    • c) Trigonometric identities
    • d) None of the above
    Answer: b) Integration by parts
69. The definite integral \( \int_0^1 (3x^2 + 2x) \, dx \) evaluates to:
    • a) 4
    • b) 3
    • c) 2
    • d) 5
    Answer: a) 4
70. The fundamental theorem of calculus states that if \( f(x) \) is continuous on the interval \( [a, b] \), then:
    • a) \( \int_a^b f(x) \, dx = f(a) + f(b) \)
    • b) \( \int_a^b f(x) \, dx = F(b) - F(a) \)
    • c) \( \int_a^b f(x) \, dx = F(a) + F(b) \)
    • d) \( \int_a^b f(x) \, dx = 0 \)
    Answer: b) \( \int_a^b f(x) \, dx = F(b) - F(a) \)
71. The area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 2 \) is:
    • a) 2
    • b) 3
    • c) 4
    • d) 5
    Answer: a) 2
72. The integral \( \int_0^\infty \frac{1}{x^2 + 1} \, dx \) evaluates to:
    • a) \( \frac{\pi}{2} \)
    • b) \( \pi \)
    • c) 1
    • d) 0
    Answer: a) \( \frac{\pi}{2} \)
73. The integration \( \int \sin(x) \cos(x) \, dx \) is simplified using:
    • a) Substitution
    • b) Trigonometric identities
    • c) Integration by parts
    • d) None of the above
    Answer: b) Trigonometric identities
74. The area between the curve \( y = x^2 \) and the x-axis from \( x = 0 \) to \( x = 1 \) is:
    • a) 1
    • b) 2
    • c) 3
    • d) 0.5
    Answer: d) 0.5
75. The integral of \( \frac{1}{x(x+1)} \) is solved using:
    • a) Substitution
    • b) Integration by parts
    • c) Partial fractions
    • d) None of the above
    Answer: c) Partial fractions

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