JEE Mathematics Properties and Applications of Definite Integrals MCQs Set 01

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

Question. \( \int_{0}^{\pi/2} \frac{f(x)}{f(x) + f\left(\frac{\pi}{2} - x\right)} dx \), where \( f(x) \neq -f\left(\frac{\pi}{2} - x\right) \) for \( 0 \leq x \leq \frac{\pi}{2} \), has the value
(a) \( f(0) \)
(b) \( f\left(\frac{\pi}{2}\right) \)
(c) \( \frac{\pi}{2} \)
(d) None of the options
Answer: (d) None of the options

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

Question. \( \int_{\pi/5}^{3\pi/10} \frac{\cos x}{\cos x + \sin x} dx \) is equal to
(a) \( \pi \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{\pi}{4} \)
(d) None of the options
Answer: (d) None of the options

Question. The equation \( \int_{-\pi/4}^{\pi/4} \left( \lambda | \sin x | + \frac{\mu \sin x}{1 + \cos x} + \nu \right) dx = 0 \), where \( \lambda, \mu, \nu \) are constants gives a relation between
(a) \( \lambda, \mu \) and \( \nu \)
(b) \( \lambda \) and \( \nu \)
(c) \( \lambda \) and \( \mu \)
(d) \( \mu \) and \( \nu \)
Answer: (b) \( \lambda \) and \( \nu \)

Question. Let \( f(x) = x - [x] \) for \( x \in \mathbb{R} \), where \( [x] = \) the greatest integer \( \leq x \). Then \( \int_{-2}^{2} f(x) dx \) is
(a) 4
(b) 2
(c) 0
(d) 1
Answer: (b) 2

Question. \( \int_{0}^{\pi} \sin^{6} x \cdot \cos^{5} x dx \) is equal to
(a) \( 2 \int_{0}^{\pi/2} \sin^{50} x \cdot \cos^{47} x dx \)
(b) 0
(c) 1
(d) None of the options
Answer: (b) 0

Question. \( \int_{-2}^{2} |1 - x^{2}| dx \) is
(a) 4
(b) 2
(c) -2
(d) 0
Answer: (a) 4

Question. \( \int_{0}^{2\pi} \frac{x \sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} dx, n > 0 \), is equal to
(a) \( \pi \)
(b) \( 2\pi \)
(c) \( \pi^{2} \)
(d) \( \frac{1}{2} \pi^{2} \)
Answer: (c) \( \pi^{2} \)

Question. \( \int_{-\pi/4}^{\pi/4} \frac{e^{x} \cdot \sec^{2} x}{e^{2x} - 1} dx \) is equal to
(a) 0
(b) 2
(c) e
(d) None of the options
Answer: (a) 0

Question. If \( [x] \) denotes the greatest integer less than or equal to \( x \) then \( \int_{0}^{\infty} \left[ \frac{2}{e^{x}} \right] dx \) is equal to
(a) \( \log_{e} 2 \)
(b) \( e^{2} \)
(c) 0
(d) \( \frac{2}{e} \)
Answer: (a) \( \log_{e} 2 \)

Question. \( \int_{1}^{3} | (2 - x) \log_{e} x | dx \) is equal to
(a) \( \frac{3}{2} \log_{e} 3 + \frac{1}{2} \)
(b) \( \log_{e} \frac{16}{3\sqrt{2}} - \frac{1}{2} \)
(c) \( -\frac{3}{2} \log_{e} 3 - \frac{1}{2} \)
(d) None of the options
Answer: (b) \( \log_{e} \frac{16}{3\sqrt{2}} - \frac{1}{2} \)

Question. \( \int_{-2}^{2} | x(x - 1) | dx \) is
(a) \( \frac{11}{3} \)
(b) \( \frac{13}{3} \)
(c) \( \frac{16}{3} \)
(d) \( \frac{17}{3} \)
Answer: (d) \( \frac{17}{3} \)

Question. The value of \( \int_{-2}^{2} \frac{\sin^{2} x}{\left[ \frac{x}{\pi} \right] + \frac{1}{2}} dx \), where \( [x] = \) the greatest integer greater than or equal to \( x \), is
(a) 1
(b) 0
(c) \( 4 - \sin 4 \)
(d) None of the options
Answer: (b) 0

Question. The value of \( \int_{0}^{\pi} [\cos x] dx \), where \( [.] \) is the greatest integer function, is
(a) \( \frac{\pi}{2} \)
(b) 0
(c) \( \pi \)
(d) \( -\frac{\pi}{2} \)
Answer: (d) \( -\frac{\pi}{2} \)

Question. Let \( a_{n} = \int_{0}^{\pi/2} \cos^{n} x \cdot \cos nx dx \). Then \( a_{n} : a_{n+1} \) is equal to
(a) 3 : 1
(b) 2 : 3
(c) 2 : 1
(d) 3 : 4
Answer: (c) 2 : 1

Question. If \( \int_{0}^{x} f(t) dt = x + \int_{x}^{1} t f(t) dt \) then the value of \( f(1) \) is
(a) \( \frac{1}{2} \)
(b) 0
(c) 1
(d) \( -\frac{1}{2} \)
Answer: (a) \( \frac{1}{2} \)

Question. The value of \( \int_{-1}^{1} \max\{2 - x, 2, 1 + x\} dx \) is
(a) 4
(b) \( \frac{9}{2} \)
(c) 2
(d) None of the options
Answer: (b) \( \frac{9}{2} \)

Question. Let \( f \) be a positive function. If \( I_{1} = \int_{1-k}^{k} xf\{x(1 - x)\} dx, I_{2} = \int_{1-k}^{k} f\{x(1 - x)\} dx \), where \( 2k - 1 > 0 \), then \( \frac{I_{1}}{I_{2}} \) is
(a) 2
(b) k
(c) \( \frac{1}{2} \)
(d) 1
Answer: (c) \( \frac{1}{2} \)

Question. If \( x \in (2n\pi, 2n\pi + \pi) \) then \( \int_{0}^{x} [\sin x] dx \), where \( [x] = \) greatest integer less than or equal to \( x \), is equal to
(a) \( -\pi \)
(b) \( -n\pi \)
(c) 0
(d) None of the options
Answer: (b) \( -n\pi \)

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

Question. \( \int_{0}^{\pi} \frac{dx}{1 + 3^{\cos x}} \) is equal to
(a) \( \pi \)
(b) 0
(c) \( \frac{\pi}{2} \)
(d) None of the options
Answer: (c) \( \frac{\pi}{2} \)

Question. If \( [y] = \) the greatest integer less than or equal to \( y \) then \( \int_{\pi/2}^{3\pi/2} [2 \sin x] dx \) is
(a) \( -\pi \)
(b) 0
(c) \( -\frac{\pi}{2} \)
(d) \( \frac{\pi}{2} \)
Answer: (c) \( -\frac{\pi}{2} \)

Question. The value of \( \int_{0}^{2\pi} \frac{dx}{e^{\sin x} + 1} \) is
(a) \( \pi \)
(b) 0
(c) \( 2\pi \)
(d) \( \frac{\pi}{2} \)
Answer: (a) \( \pi \)

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

Question. \( \int_{0}^{\pi/4} \sin x d(x - [x]) \) is equal to
(a) \( \frac{1}{2} \)
(b) \( 1 - \frac{1}{\sqrt{2}} \)
(c) 1
(d) None of the options
Answer: (b) \( 1 - \frac{1}{\sqrt{2}} \)

Question. If \( f(x) = \int_{0}^{\sin x} \cos^{-1} t dt + \int_{0}^{\cos x} \sin^{-1} t dt \), \( 0 < x < \frac{\pi}{2} \), then \( f\left(\frac{\pi}{4}\right) \) is
(a) \( \frac{\pi}{\sqrt{2}} \)
(b) \( 1 + \frac{\pi}{2\sqrt{2}} \)
(c) 1
(d) None of the options
Answer: (b) \( 1 + \frac{\pi}{2\sqrt{2}} \)

Question. The value of \( \int_{\alpha}^{\beta} x | x | dx \), where \( \alpha < 0 < \beta \), is
(a) \( \frac{1}{2} (\alpha^{2} + \beta^{2}) \)
(b) \( \frac{1}{3} (\beta^{2} - \alpha^{2}) \)
(c) \( \frac{1}{3} (\alpha^{2} + \beta^{2}) \)
(d) None of the options
Answer: (c) \( \frac{1}{3} (\alpha^{2} + \beta^{2}) \)

Question. The value of \( \int_{0}^{\pi/4} \log(1 + \tan x) dx \) is equal to
(a) \( \frac{\pi}{8} \log_{e} 2 \)
(b) \( \frac{\pi}{4} \log_{e} 2 \)
(c) \( \frac{\pi}{4} \)
(d) None of the options
Answer: (a) \( \frac{\pi}{8} \log_{e} 2 \)

Question. \( \int_{-1}^{1} (x - [2x]) dx \) is equal to
(a) 1
(b) 0
(c) 2
(d) 4
Answer: (a) 1

Question. If \( f(x) = |x| + 1, -1 \leq x < 0 \)
\( 1 + |x|^{2}, 0 \leq x \leq 1 \)
then \( \int_{-1}^{1} f(x) dx \) is equal to
(a) \( -\frac{1}{6} \)
(b) \( \frac{17}{6} \)
(c) \( -\frac{17}{6} \)
(d) None of the options
Answer: (b) \( \frac{17}{6} \)

Question. Let \( f(x) = \max \{x + |x|, - [x]\} \), where \( [x] = \) the greatest integer \( \leq x \). Then \( \int_{-2}^{2} f(x) dx \) is equal to
(a) 3
(b) 2
(c) 1
(d) None of the options
Answer: (d) None of the options

Question. \( \int_{0}^{3} | x^{3} - 3x^{2} + 2x | dx \) is equal to
(a) \( \frac{3}{4} \)
(b) \( \frac{7}{4} \)
(c) \( \frac{11}{4} \)
(d) None of the options
Answer: (c) \( \frac{11}{4} \)

Question. Let \( f(x) \) be a continuous function such that \( f(a - x) + f(x) = 0 \) for all \( x \in [0, a] \). Then \( \int_{0}^{a} \frac{dx}{1 + e^{f(x)}} \) is equal to
(a) a
(b) \( \frac{a}{2} \)
(c) f(a)
(d) \( \frac{1}{2} f(a) \)
Answer: (b) \( \frac{a}{2} \)

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

Question. If \( A = \int_{0}^{\pi} \frac{\sin x}{\sin x + \cos x} dx, B = \int_{0}^{\pi} \frac{\sin x}{\sin x - \cos x} dx \) then
(a) A + B = 0
(b) A = B
(c) A = B = \( \pi/2 \)
(d) A = -B = \( \pi \)
Answer: (b) A = B
(c) A = B = \( \pi/2 \)

Question. Let \( f(a) > 0 \), and let \( f(x) \) be a nondecreasing continuous function in [a, b]. Then \( \frac{1}{b - a} \int_{a}^{b} f(x) dx \) has the
(a) maximum value of f(b)
(b) minimum value f(a)
(c) maximum value bf(b)
(d) minimum value \( \frac{f(a)}{b - a} \)
Answer: (a) maximum value of f(b)
(b) minimum value f(a)

Question. The value of \( \int_{0}^{\pi} \frac{\sin nx}{\sin x} dx, n \in \mathbb{N} \), is
(a) \( \pi \) if n is even
(b) 0 if n is odd
(c) 0 if n is even
(d) \( \pi \) for all \( n \in \mathbb{N} \)
Answer: (c) 0 if n is even