JEE Mathematics Definite Integration MCQs Set 04

Question. Let \( l_1 = \int_{0}^{1} \frac{e^x dx}{1 + x} \) and \( I_2 = \int_{0}^{1} \frac{x^2 dx}{e^{x^3}(2 - x^3)} \), then \( \frac{l_1}{I_2} \) is to
(a) 3/e
(b) e/3
(c) 3e
(d) 1/3e
Answer: (c) 3e

Question. If f(x) is a continuous function and attains only rational values in \( [-3, 3] \) and its greatest value in \( [-3, 3] \) is 5, then \( \int_{-3}^{3} f(x) dx = \)
(a) 5
(b) 10
(c) 20
(d) 30
Answer: (d) 30

Question. Let \( f(x) = \text{minimum } (|x|, 1 - |x|, 1/4), \ \forall \ x \in R \), then the value of \( \int_{-1}^{1} f(x) dx \) is equal to
(a) \( \frac{1}{32} \)
(b) \( \frac{3}{8} \)
(c) \( \frac{4}{32} \)
(d) None of the options
Answer: (b) \( \frac{3}{8} \)

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

Question. Let \( f(x) = \int_{0}^{x} (t^2 - t + 1) dt \ \forall \ x \in (3, 4) \), then the difference between the greatest and the least values of the function is
(a) \( \frac{49}{6} \)
(b) \( \frac{59}{6} \)
(c) \( \frac{69}{8} \)
(d) \( \frac{59}{3} \)
Answer: (b) \( \frac{59}{6} \)

Question. For \( 0 < x < \frac{\pi}{2} \), \( \int_{1/\sqrt{2}}^{1/2} \cot x \ d(\cos x) \) equals to
(a) \( \frac{\sqrt{3} - \sqrt{2}}{2} \)
(b) \( \frac{\sqrt{2} - \sqrt{3}}{2} \)
(c) \( \frac{1 - \sqrt{3}}{2} \)
(d) None of the options
Answer: (b) \( \frac{\sqrt{2} - \sqrt{3}}{2} \)

Question. If \( f(x) = \begin{cases} e^{\cos x} \sin x & , |x| \le 2 \\ 2 & , \text{otherwise} \end{cases} \), then \( \int_{-2}^{3} f(x) dx = \)
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (c) 2

Question. The value of \( \int_{0}^{\pi/3} [\sqrt{3} \tan x] dx \)
(where \( [ \ \ ] \) denotes the greatest integer function)
(a) \( \frac{5\pi}{6} \)
(b) \( \frac{5\pi}{6} - \tan^{-1} \left( \frac{2}{\sqrt{3}} \right) \)
(c) \( \frac{\pi}{2} - \tan^{-1} \left( \frac{2}{\sqrt{3}} \right) \)
(d) None of the options
Answer: (c) \( \frac{\pi}{2} - \tan^{-1} \left( \frac{2}{\sqrt{3}} \right) \)

Question. \( \int_{-1}^{1} \frac{\sin x + x^2}{3 - |x|} dx \)
(a) 0
(b) \( 2 \int_{0}^{1} \frac{\sin x}{3 - |x|} dx \)
(c) \( 2 \int_{0}^{1} \frac{x^2}{3 - |x|} dx \)
(d) \( 2 \int_{0}^{1} \frac{\sin x + x^2}{3 - |x|} dx \)
Answer: (c) \( 2 \int_{0}^{1} \frac{x^2}{3 - |x|} dx \)

Question. Let \( I_1 = \int_{1}^{2} \frac{dx}{\sqrt{1 + x^2}} \) and \( I_2 = \int_{1}^{2} \frac{dx}{x} \)
(a) \( I_1 > I_2 \)
(b) \( I_2 > I_1 \)
(c) \( l_1 = I_2 \)
(d) \( I_1 > 2I_2 \)
Answer: (b) \( I_2 > I_1 \)

Question. \( \int_{5/2}^{5} \frac{\sqrt{(25 - x^2)^3}}{x^4} dx \) equals to
(a) \( \frac{\pi}{3} \)
(b) \( \frac{2\pi}{3} \)
(c) \( \frac{\pi}{6} \)
(d) None of the options
Answer: (a) \( \frac{\pi}{3} \)

Question. The value of \( \int_{-2}^{1} \left[ x \left[ 1 + \cos \left( \frac{\pi x}{2} \right) \right] + 1 \right] dx \) is
(where \( [ \ \ ] \) denotes the greatest integer function)
(a) 1
(b) 1/2
(c) 2
(d) None of the options
Answer: (c) 2

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

Question. If \( x \in (0, 2) \) then the value of \( \int_{0}^{1} e^{2x - [2x]} d(x - [x]) \) is
(where \( [ \ \ ] \) denotes the greatest integer function)
(a) e + 1
(b) e
(c) 2e - 2
(d) None of the options
Answer: (d) None of the options

Question. \( \lim_{n \to \infty} \sum_{r=1}^{n} \frac{\sqrt{n}}{\sqrt{r} (3\sqrt{r} + 4\sqrt{n})^2} = \)
(a) \( \frac{1}{7} \)
(b) \( \frac{1}{10} \)
(c) \( \frac{1}{14} \)
(d) None of the options
Answer: (c) \( \frac{1}{14} \)

Question. The value of \( \int_{\pi/4}^{\pi/3} \operatorname{cosec} x \ d(\sin x) \) for \( 0 < x < \pi/2 \) is
(a) \( \ln 2 \)
(b) \( \frac{1}{2} \ln \frac{3}{2} \)
(c) \( \ln \left( \frac{\sin 1/2}{\sin 1/\sqrt{2}} \right) \)
(d) None of the options
Answer: (d) None of the options

Question. \( \int_{0}^{2} x^3 \left[ 1 + \cos \frac{\pi x}{2} \right] dx \) is
(where \( [ \ \ ] \) denotes the greatest integer function)
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{4} \)
(c) 0
(d) None of the options
Answer: (b) \( \frac{1}{4} \)

Question. If \( I = \int_{0}^{2\pi} \sin^2 x \,dx \), then
(a) \( I = 2 \int_{0}^{\pi} \sin^2 x \,dx \)
(b) \( I = 4 \int_{0}^{\pi/2} \sin^2 x \,dx \)
(c) \( I = \int_{0}^{2\pi} \cos^2 x \,dx \)
(d) \( I = 8 \int_{0}^{\pi/4} \sin^2 x \,dx \)
Answer: (a) \( I = 2 \int_{0}^{\pi} \sin^2 x \,dx \), (b) \( I = 4 \int_{0}^{\pi/2} \sin^2 x \,dx \), (c) \( I = \int_{0}^{2\pi} \cos^2 x \,dx \)