JEE Mathematics Properties and Applications of Definite Integrals MCQs Set 02

Choose the most appropriate option (a, b, c or d).

Question. \( \int_{-a}^{a} \log_{e} (x + \sqrt{1 + x^{2}}) dx \) is equal to
(a) \( 2 \log_{e} a \)
(b) 0
(c) \( \log_{e} 2 + \log a \)
(d) None of the options
Answer: (b) 0

Question. Let \( f(x) = \frac{e^{x} + 1}{e^{x} - 1} \) and \( \int_{0}^{1} \frac{e^{x} + 1}{e^{x} - 1} \cdot x dx = \lambda \). Then \( \int_{-1}^{1} t f(t) dt \) is equal to
(a) 0
(b) \( 2\lambda \)
(c) \( \lambda \)
(d) None of the options
Answer: (b) \( 2\lambda \)

Question. Let \( f(x) \) be a continuous function in \( \mathbb{R} \) such that \( f(x) + f(y) = f(x + y) \). If \( \int_{0}^{3} f(x) dx = k \) then \( \int_{-3}^{3} f(x) dx \) is equal to
(a) 2k
(b) 0
(c) k/2
(d) -2k
Answer: (b) 0

Question. Let \( f(x) \) be a continuous function such that \( f(x) \) does not vanish for all \( x \in \mathbb{R} \). If \( \int_{-2}^{3} f(x) dx = \int_{-2}^{3} f(x) dx \) then \( f(x), x \in \mathbb{R} \), is
(a) an even function
(b) an odd function
(c) a periodic function
(d) None of the options
Answer: (d) None of the options

Question. Let \( I = \int_{-a}^{a} (p \tan^{3} x + q \cos^{2} x + r \sin x) dx \), where \( p, q, r \) are arbitrary constants. The numerical value of \( I \) depends on
(a) p, q, r
(b) q, r, a
(c) q, a
(d) p, r, a
Answer: (c) q, a

Question. Let \( I_{1} = \int_{0}^{1} e^{-x^{2}} dx \), \( I_{2} = \int_{0}^{1} e^{-x^{2}} \cos^{2} x dx \) and \( I_{3} = \int_{0}^{1} e^{-x^{2}} \cos^{2} x dx \). Then
(a) \( I_{1} < I_{2} < I_{3} \)
(b) \( I_{3} < I_{2} < I_{1} \)
(c) \( I_{2} < I_{1} < I_{3} \)
(d) \( I_{2} < I_{3} < I_{1} \)
Answer: (d) \( I_{2} < I_{3} < I_{1} \)

Question. \( \int_{0}^{100\pi} \sqrt{1 + \cos 2x} dx \) is equal to
(a) 0
(b) \( 100\sqrt{2} \)
(c) \( 200\sqrt{2} \)
(d) 100
Answer: (c) \( 200\sqrt{2} \)

Question. The value of \( \int_{0}^{1} (1 + e^{-x^{2}}) dx \) is
(a) -1
(b) 2
(c) \( 1 + e^{-1} \)
(d) None of the options
Answer: (d) None of the options

Question. If \( \int_{-2}^{3} f(x) dx = 5 \) and \( \int_{1}^{3} \{2 - f(x)\} dx = 6 \) then the value of \( \int_{-2}^{1} f(x) dx \) is
(a) 7
(b) 3
(c) -7
(d) -3
Answer: (a) 7

Question. Let \( f(x) \) be a continuous function such that \( \int_{n}^{n+1} f(x) dx = n^{3}, n \in \mathbb{Z} \). Then the value of \( \int_{-3}^{3} f(x) dx \) is
(a) 9
(b) -27
(c) -9
(d) None of the options
Answer: (b) -27

Question. If \( I = \int_{0}^{1} \frac{x dx}{8 + x^{2}} \) then the smallest interval in which I lies is
(a) \( (0, 1/8) \)
(b) \( (0, 1/9) \)
(c) \( (0, 1/10) \)
(d) \( (0, 1/7) \)
Answer: (b) \( (0, 1/9) \)

Question. If \( \int_{0}^{1} xe^{x^{2}} dx = \lambda \int_{0}^{1} e^{x^{2}} dx \) then
(a) \( \lambda = 0 \)
(b) \( \lambda \in (0, 1) \)
(c) \( \lambda \in (-\infty, 0) \)
(d) \( \lambda \in (1, 2) \)
Answer: (b) \( \lambda \in (0, 1) \)

Question. \( \int_{\log 1/2}^{\log 2} \sin \left( \frac{e^{x} - 1}{e^{x} + 1} \right) dx \) is equal to
(a) \( \cos \frac{1}{3} \)
(b) 0
(c) 2 cos 2
(d) None of the options
Answer: (b) 0

Question. Let \( \int_{0}^{a} f(x) dx = \lambda \) and \( \int_{0}^{a} f(2a - x) dx = \mu \). Then \( \int_{0}^{2a} f(x) dx \) is equal to
(a) \( \lambda + \mu \)
(b) \( \lambda - \mu \)
(c) \( 2\lambda - \mu \)
(d) \( \lambda - 2\mu \)
Answer: (a) \( \lambda + \mu \)

Question. If \( f(-x) + f(x) = 0 \) then \( \int_{a}^{x} f(t) dt \) is
(a) an odd function
(b) an even function
(c) a periodic function
(d) None of the options
Answer: (b) an even function

Question. If \( f(x) \) and \( g(x) \) be continuous functions over the closed interval [0, a] such that \( f(x) = f(a - x) \) and \( g(x) + g(a - x) = 2 \). Then \( \int_{0}^{a} f(x) \cdot g(x) dx \) is equal to
(a) \( \int_{0}^{a} f(x) dx \)
(b) \( \int_{0}^{a} g(x) dx \)
(c) 2a
(d) None of the options
Answer: (a) \( \int_{0}^{a} f(x) dx \)

Question. If \( f(x) = f(a + x) \) and \( \int_{0}^{a} f(x) dx = p \) then \( \int_{a}^{na} f(x) dx \) is equal to
(a) np
(b) (n - 1)p
(c) (n + 1)p
(d) None of the options
Answer: (b) (n - 1)p

Question. Let \( f(x) \) be a given integrable function such that \( f(x + k) = f(x) \) for all \( x \in \mathbb{R} \). Then \( \int_{a}^{a+k} f(x) dx \) depends for its value on
(a) a only
(b) k only
(c) both a and k
(d) neither a nor k
Answer: (b) k only

Question. The value of \( \int_{\pi/4}^{3\pi/4} \frac{x}{1 + \sin x} dx \) is equal to
(a) \( (\sqrt{2} - 1)\pi \)
(b) \( (\sqrt{2} + 1)\pi \)
(c) \( \pi \)
(d) None of the options
Answer: (a) \( (\sqrt{2} - 1)\pi \)

Question. The value of \( \int_{0}^{\pi/2} \frac{dx}{1 + \tan^{3} x} \) is
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{4} \)
(c) \( \pi \)
(d) None of the options
Answer: (b) \( \frac{\pi}{4} \)

Question. Let \( f \) and \( g \) be two continuous functions. Then \( \int_{-\pi/2}^{\pi/2} \{f(x) + f(-x)\}\{g(x) - g(-x)\} dx \) is equal to
(a) \( \pi \)
(b) 1
(c) -1
(d) 0
Answer: (d) 0

Question. Let \( \int_{a}^{b} f(x) dx = p \) and \( \int_{a}^{b} | f(x) | dx = q \). Then
(a) \( |p| \leq q \)
(b) \( p > q \)
(c) \( p + q = 0 \)
(d) None of the options
Answer: (a) \( |p| \leq q \)

Question. If \( f(x) = \int_{0}^{x} \log(1 + t^{2}) dt \) then the value of \( f''(1) \) is equal to
(a) 2
(b) 0
(c) 1
(d) None of the options
Answer: (c) 1

Question. If \( f(x) = \int_{x}^{x^2} \frac{dt}{1 + t^{3}} \) then \( f'(2) \) is equal to
(a) \( \frac{101}{585} \)
(b) \( -\frac{29}{585} \)
(c) \( -\frac{56}{585} \)
(d) None of the options
Answer: (b) \( -\frac{29}{585} \)

Question. If \( f(2a - x) = f(x) \) and \( \int_{0}^{a} f(x) dx = \lambda \) then \( \int_{0}^{2a} f(x) dx \) is
(a) \( 2\lambda \)
(b) \( \lambda \)
(c) 0
(d) None of the options
Answer: (a) \( 2\lambda \)

Question. Let \( f(x) \) be a continuous function such that the area bounded by the curve \( y = f(x) \), the x-axis, and the lies \( x = 0 \) and \( x = a \) is \( 1 + \frac{a^{2}}{2} \sin a \). Then
(a) \( f\left(\frac{\pi}{2}\right) = 1 + \frac{\pi^{2}}{8} \)
(b) \( f(a) = 1 + \frac{a^{2}}{2} \sin a \)
(c) \( f(a) = a \sin a + \frac{1}{2} a^{2} \cos a \)
(d) None of the options
Answer: (c) \( f(a) = a \sin a + \frac{1}{2} a^{2} \cos a \)

Question. If \( y = \int_{0}^{x} \sqrt{\sin x} dx \) then the value of \( \frac{dy}{dx} \) at \( x = \frac{\pi}{2} \) is
(a) 0
(b) 1
(c) -1
(d) None of the options
Answer: (b) 1

Question. If \( y = \int_{x}^{x^2} \sqrt{5 - t^{2}} dt \) then the value of \( \frac{dy}{dx} \) at \( x = \sqrt{2} \) is
(a) \( 1 - \sqrt{3} \)
(b) \( \sqrt{3}(2\sqrt{6} - 1) \)
(c) \( 2\sqrt{2} - \sqrt{3} \)
(d) None of the options
Answer: (c) \( 2\sqrt{2} - \sqrt{3} \)

Question. The function \( f(x) = \int_{1}^{x} t(e^{t} - 1)(t - 2)^{3}(t - 3)^{5} dt \) has a local minimum at x which is equal to
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (d) 3

Question. Let \( f(x) \) be a differentiable function and \( f(1) = 2 \). If \( \lim_{x \to 1} \frac{\int_{2}^{f(x)} 2t dt}{x - 1} = 4 \) then the value of \( f'(1) \) is
(a) 1
(b) 2
(c) 4
(d) None of the options
Answer: (a) 1

Question. If \( \phi(x) = \int_{x}^{x^2} (t - 1) dt \), \( 1 \leq x \leq 2 \), then the greatest value of \( \phi(x) \) is
(a) 2
(b) 4
(c) 8
(d) None of the options
Answer: (b) 4

Question. If \( \int_{0}^{1} (1 + \sin^{4} x)(ax^{2} + bx + c) dx = \int_{0}^{2} (1 + \sin^{4} x)(ax^{2} + bx + c) dx \) then the quadratic equation \( ax^{2} + bx + c = 0 \) has
(a) at least one root in (1, 2)
(b) no root in (1, 2)
(c) two equal roots in (1, 2)
(d) both roots imaginary
Answer: (a) at least one root in (1, 2)

Question. Let \( f(x) \) be a function defined by \( f(x) = \int_{1}^{x} x(x^{2} - 3x + 2) dx \), \( 1 \leq x \leq 3 \). Then the range of \( f(x) \) is
(a) [0, 2]
(b) \( [-\frac{1}{4}, 4] \)
(c) \( [-\frac{1}{4}, 2] \)
(d) None of the options
Answer: (c) \( [-\frac{1}{4}, 2] \)

Choose the correct options. One or more options may be correct.

Question. If \( f(x) = \int_{x}^{x^2} \frac{dt}{(\log t)^{2}}, x \neq 0, x \neq 1 \), then \( f(x) \) is
(a) monotonically increasing in (2, \( +\infty \))
(b) monotonically increasing in (1, 2)
(c) monotonically increasing in (2, \( +\infty \))
(d) monotonically decreasing in (0, 1)
Answer: (a) monotonically increasing in (2, \( +\infty \))
(d) monotonically decreasing in (0, 1)

Question. Let \( f(x) = ax^{3} + bx^{2} + cx \) have relative extrema at \( x = 1 \) and at \( x = 5 \). If \( \int_{-1}^{1} f(x) dx = 6 \) then
(a) a = -1
(b) b = 9
(c) c = 15
(d) a = 1
Answer: (a) a = -1
(b) b = 9

Question. Let \( f(x) = \int_{0}^{x} | x - 1 | dx \), \( x \geq 0 \). Then \( f'(x) \) is
(a) continuous at x = 1
(b) continuous at x = 2
(c) differentiable at x = 1
(d) differentiable at x = 2
Answer: (a) continuous at x = 1
(b) continuous at x = 2
(d) differentiable at x = 2