JEE (Main) Mathematics - Integral Calculus MCQs (31 to 45)

31. Question 31: The Fundamental Theorem of Calculus states that:

    • a) \( \int_{a}^{b} f'(x) dx = f(b) - f(a) \)
    • b) \( \int_{a}^{b} f(x) dx = f'(b) - f'(a) \)
    • c) \( \int_{a}^{b} f(x) dx = f(a) - f(b) \)
    • d) None of the above

    Answer: a) \( \int_{a}^{b} f'(x) dx = f(b) - f(a) \)

32. Question 32: Which of the following is a property of definite integrals?

    • a) \( \int_{a}^{b} f(x) dx = \int_{b}^{a} f(x) dx \)
    • b) \( \int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx \)
    • c) \( \int_{a}^{b} f(x) dx = 0 \)
    • d) None of the above

    Answer: b) \( \int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx \)

33. Question 33: The value of the integral \( \int_{0}^{\pi} \sin(x) \, dx \) is:

    • a) 1
    • b) 2
    • c) 0
    • d) -1

    Answer: b) 2

34. Question 34: The area of the region bounded by the curve \( y = x^2 \), the x-axis, and the lines \( x = 0 \) and \( x = 2 \) is:

    • a) 2
    • b) 4
    • c) 8
    • d) 3

    Answer: b) 4

35. Question 35: If \( F(x) \) is an antiderivative of \( f(x) \), then the value of \( \int_{a}^{b} f(x) \, dx \) is:

    • a) \( F(b) - F(a) \)
    • b) \( F(a) - F(b) \)
    • c) \( F(a) + F(b) \)
    • d) None of the above

    Answer: a) \( F(b) - F(a) \)

36. Question 36: The area between the curve \( y = x^2 \) and the x-axis from \( x = -1 \) to \( x = 1 \) is:

    • a) 1
    • b) 2
    • c) 0
    • d) 4

    Answer: b) 2

37. Question 37: The integral \( \int_{0}^{1} (x^3 + 2x) \, dx \) is evaluated as:

    • a) \( \frac{1}{4} \)
    • b) 1
    • c) \( \frac{3}{4} \)
    • d) \( \frac{5}{4} \)

    Answer: d) \( \frac{5}{4} \)

38. Question 38: The area of the region bounded by the curve \( y = \sin(x) \), the x-axis, and the lines \( x = 0 \) and \( x = \pi \) is:

    • a) 0
    • b) 1
    • c) 2
    • d) \( \pi \)

    Answer: b) 2

39. Question 39: Which of the following integrals represents the area under the curve \( y = \ln(x) \) from \( x = 1 \) to \( x = e \)?

    • a) \( \int_{1}^{e} \ln(x) \, dx \)
    • b) \( \int_{0}^{e} \ln(x) \, dx \)
    • c) \( \int_{1}^{e} x \ln(x) \, dx \)
    • d) \( \int_{1}^{e} \frac{\ln(x)}{x} \, dx \)

    Answer: a) \( \int_{1}^{e} \ln(x) \, dx \)

40. Question 40: The area of the region bounded by the curve \( y = x^2 - 4x \), the x-axis, and the lines \( x = 0 \) and \( x = 4 \) is:

    • a) 8
    • b) 16
    • c) 12
    • d) 0

    Answer: a) 8

41. Question 41: The integral \( \int_{0}^{1} \frac{1}{1 + x^2} \, dx \) gives:

    • a) 0
    • b) 1
    • c) \( \frac{\pi}{4} \)
    • d) \( \frac{\pi}{2} \)

    Answer: c) \( \frac{\pi}{4} \)

42. Question 42: Which of the following represents the area under the curve \( y = x^3 \) between \( x = 0 \) and \( x = 2 \)?

    • a) \( \int_{0}^{2} x^3 \, dx \)
    • b) \( \int_{0}^{2} 3x^2 \, dx \)
    • c) \( \int_{0}^{2} 2x^3 \, dx \)
    • d) \( \int_{0}^{2} x^2 \, dx \)

    Answer: a) \( \int_{0}^{2} x^3 \, dx \)

43. Question 43: The integral \( \int_{0}^{1} (x + 2) \, dx \) is:

    • a) 1
    • b) 2
    • c) 3
    • d) 4

    Answer: c) 3

44. Question 44: The area of the region bounded by the curve \( y = x^3 \), the x-axis, and the lines \( x = -1 \) and \( x = 1 \) is:

    • a) 0
    • b) 1
    • c) 2
    • d) 4

    Answer: a) 0

45. Question 45: The integral \( \int_{-1}^{1} e^{-x^2} \, dx \) is known to be:

    • a) \( \sqrt{\pi} \)
    • b) 0
    • c) 1
    • d) 2

    Answer: a) \( \sqrt{\pi} \)

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