JEE Mathematics Definite Integration MCQs Set 02

Question. If \( \int_{1}^{x} \frac{dt}{|t| \sqrt{t^2 - 1}} = \frac{\pi}{6} \), then x can be equal to
(a) \( \frac{2}{\sqrt{3}} \)
(b) \( \sqrt{3} \)
(c) 2
(d) None of the options
Answer: (a) \( \frac{2}{\sqrt{3}} \)

Question. If \( f(x) = \begin{cases} x & ; x < 1 \\ x - 1 & ; x \ge 1 \end{cases} \), then \( \int_{0}^{2} x^2 f(x) dx \) is equal to
(a) 1
(b) \( \frac{4}{3} \)
(c) \( \frac{5}{3} \)
(d) \( \frac{5}{2} \)
Answer: (c) \( \frac{5}{3} \)

Question. Suppose for every integer n, \( \int_{n}^{n+1} f(x) dx = n^2 \). The value of \( \int_{-2}^{4} f(x) dx \) is
(a) 16
(b) 14
(c) 19
(d) 21
Answer: (c) 19

Question. \( \int_{0}^{\pi} |1 + 2 \cos x| dx \) equals to :
(a) \( \frac{2\pi}{3} \)
(b) \( \pi \)
(c) 2
(d) \( \frac{\pi}{3} + 2\sqrt{3} \)
Answer: (d) \( \frac{\pi}{3} + 2\sqrt{3} \)

Question. The value of \( \int_{-1}^{3} (|x - 2| + [x]) dx \) is equal to
(where \( [ \ \ ] \) denotes greatest integer function)
(a) 7
(b) 5
(c) 4
(d) 3
Answer: (a) 7

Question. Let \( f : R \to R, g : R \to R \) be continuous functions. Then the value of integral \( \int_{\ln \lambda}^{\ln 1/\lambda} \frac{f \left( \frac{x^2}{4} \right) [f(x) - f(-x)]}{g \left( \frac{x^2}{4} \right) [g(x) + g(-x)]} dx \) is
(a) depend on \( \lambda \)
(b) a non-zero constant
(c) zero
(d) None of the options
Answer: (c) zero

Question. If \( \int_{-1}^{3/2} |x \sin \pi x| dx = \frac{k}{\pi^2} \), then the value of k is
(a) \( 3\pi + 1 \)
(b) \( 2\pi + 1 \)
(c) 1
(d) 4
Answer: (a) \( 3\pi + 1 \)

Question. \( \int_{0}^{\pi/4} \frac{x \sin x}{\cos^3 x} dx \) equals to :
(a) \( \frac{\pi}{4} + \frac{1}{2} \)
(b) \( \frac{\pi}{4} - \frac{1}{2} \)
(c) \( \frac{\pi}{4} \)
(d) None of the options
Answer: (b) \( \frac{\pi}{4} - \frac{1}{2} \)

Question. If f(0) = 1, f(2) = 3, f '(2) = 5 and f '(0) is finite, then \( \int_{0}^{1} x \cdot f''(2x) dx \) is equal to
(a) zero
(b) 1
(c) 2
(d) None of the options
Answer: (c) 2

Question. \( \int_{\log \pi - \log 2}^{\log \pi} \frac{e^x}{1 - \cos \left( \frac{2}{3} e^x \right)} dx \) is equal to
(a) \( \sqrt{3} \)
(b) \( -\sqrt{3} \)
(c) \( \frac{1}{\sqrt{3}} \)
(d) \( -\frac{1}{\sqrt{3}} \)
Answer: (a) \( \sqrt{3} \)

Question. If \( I_1 = \int_{e}^{e^2} \frac{dx}{\ln x} \) and \( I_2 = \int_{1}^{2} \frac{e^x}{x} dx \), then
(a) \( I_1 = I_2 \)
(b) \( 2 I_1 = I_2 \)
(c) \( I_1 = 2 I_2 \)
(d) None of the options
Answer: (a) \( I_1 = I_2 \)

Question. \( \int_{2 - \log 3}^{3 + \log 3} \frac{\log(4 + x)}{\log(4 + x) + \log(9 - x)} dx \)
(a) cannot be evaluated
(b) is equal to \( \frac{5}{2} \)
(c) is equal to \( 1+2 \log 3 \)
(d) is equal to \( \frac{1}{2} + \log 3 \)
Answer: (d) is equal to \( \frac{1}{2} + \log 3 \)

Question. Let \( I_1 = \int_{0}^{3\pi} f(\cos^2 x) dx \), \( I_2 = \int_{0}^{2\pi} f(\cos^2 x) dx \) and \( I_3 = \int_{0}^{\pi} f(\cos^2 x) dx \), then
(a) \( I_1 + 2I_3 = 3I_2 \)
(b) \( I_1 = 2I_2 + I_3 \)
(c) \( I_2 + I_3 = I_1 \)
(d) \( I_1 = 2I_3 \)
Answer: (c) \( I_2 + I_3 = I_1 \)

Question. If \( \int_{0}^{11} \frac{11^x}{11^{[x]}} dx = \frac{k}{\log 11} \) then value of k is
(where \( [ \ \ ] \) denotes greatest integer function)
(a) 11
(b) 101
(c) 110
(d) None of the options
Answer: (c) 110

Question. The value of function \( f(x) = 1 + x + \int_{1}^{x} (\ln^2 t + 2\ln t) dt \) where \( f '(x) \) vanishes is
(a) \( e^{-1} \)
(b) 0
(c) \( 2e^{-1} \)
(d) \( 1 + 2e^{-1} \)
Answer: (d) \( 1 + 2e^{-1} \)

Question. If \( \int_{a}^{y} \cos t^2 dt = \int_{a}^{x^2} \frac{\sin t}{t} dt \), then the value of \( \frac{dy}{dx} \) is
(a) \( \frac{2 \sin^2 x}{x \cos^2 y} \)
(b) \( \frac{2 \sin x^2}{x \cos y^2} \)
(c) \( \frac{2 \sin x^2}{x \left(1 - 2 \sin^2 \frac{y^2}{2}\right)} \)
(d) None of the options
Answer: (b) \( \frac{2 \sin x^2}{x \cos y^2} \)

Question. \( \lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{r^3}{r^4 + n^4} \right) \) equals
(a) \( \log 2 \)
(b) \( \frac{1}{2} \log 2 \)
(c) \( \frac{1}{3} \log 2 \)
(d) \( \frac{1}{4} \log 2 \)
Answer: (d) \( \frac{1}{4} \log 2 \)

Question. \( \lim_{n \to \infty} \sum_{r=2n+1}^{3n} \frac{n}{r^2 - n^2} \) is equal to
(a) \( \log \sqrt{\frac{2}{3}} \)
(b) \( \log \sqrt{\frac{3}{2}} \)
(c) \( \log \frac{2}{3} \)
(d) \( \log \frac{3}{2} \)
Answer: (b) \( \log \sqrt{\frac{3}{2}} \)

Question. The value of \( \lim_{n \to \infty} \left[ \left(1 + \frac{1}{n^2}\right) \left(1 + \frac{2^2}{n^2}\right) \dots \left(1 + \frac{n^2}{n^2}\right) \right]^{1/n} \) is
(a) \( \frac{e^{\pi/2}}{2e^2} \)
(b) \( 2e^2 e^{\pi/2} \)
(c) \( \frac{2}{e^2} e^{\pi/2} \)
(d) None of the options
Answer: (c) \( \frac{2}{e^2} e^{\pi/2} \)

Question. \( \lim_{n \to \infty} \frac{\pi}{n} \left[ \sin \frac{\pi}{n} + \sin \frac{2\pi}{n} + \dots + \sin \frac{(n-1)\pi}{n} \right] \) equals
(a) 0
(b) \( \pi \)
(c) 2
(d) None of the options
Answer: (c) 2

Question. If f(x) is a function satisfying \( f \left( \frac{1}{x} \right) + x^2 f(x) = 0 \) for all non-zero x, then \( \int_{\sin \theta}^{\operatorname{cosec} \theta} f(x) dx \) equals
(a) \( \sin \theta + \operatorname{cosec} \theta \)
(b) \( \sin^2 \theta \)
(c) \( \operatorname{cosec}^2 \theta \)
(d) None of the options
Answer: (d) None of the options

Question. \( \int_{0}^{(\pi/2)^{1/3}} x^5 \cdot \sin x^3 dx \) equals to
(a) 1
(b) 1/2
(c) 2
(d) 1/3
Answer: (d) 1/3

Question. \( \lim_{n \to \infty} \left( \sin \frac{\pi}{2n} \cdot \sin \frac{2\pi}{2n} \cdot \sin \frac{3\pi}{2n} \dots \sin \frac{(n-1)\pi}{2n} \right)^{1/n} \) is equal to
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{4} \)
(d) None of the options
Answer: (a) \( \frac{1}{2} \)

Question. If f(x) and g(x) are continuous functions satisfying \( f(x) = f(a - x) \) and \( g(x) + g(a - x) = 2 \), then \( \int_{0}^{a} f(x) g(x) dx \) is equal to
(a) \( \int_{0}^{a} g(x) dx \)
(b) \( \int_{0}^{a} f(x) dx \)
(c) 0
(d) None of the options
Answer: (b) \( \int_{0}^{a} f(x) dx \)

Question. If [x] stands for the greatest integer function, the value of \( \int_{4}^{10} \frac{[x^2]}{[x^2 - 28x + 196] + [x^2]} dx \) is
(a) 0
(b) 1
(c) 3
(d) None of the options
Answer: (c) 3

Question. \( \int_{0}^{\infty} [2 e^{-x}] dx \) is equal to
(where \( [ \ \ ] \) denotes the greatest integer function)
(a) 0
(b) \( \ln 2 \)
(c) \( e^2 \)
(d) \( 2e^{-1} \)
Answer: (b) \( \ln 2 \)

Question. If \( \int_{0}^{100} f(x) dx = a \), then \( \sum_{r=1}^{100} \left( \int_{0}^{1} f(r - 1 + x) dx \right) = \)
(a) 100 a
(b) a
(c) 0
(d) 10 a
Answer: (b) a

Question. If \( f(x) = \int_{0}^{x} \sin [2x] dx \) then \( f(\pi/2) \) is
(where \( [ \ \ ] \) denotes greatest integer function)
(a) \( \frac{1}{2} \{ \sin 1 + (\pi - 2) \sin 2 \} \)
(b) \( \frac{1}{2} \{ \sin 1 + \sin 2 + (\pi - 3) \sin 3 \} \)
(c) 0
(d) \( \sin 1 + \left( \frac{\pi}{2} - 2 \right) \sin 2 \)
Answer: (b) \( \frac{1}{2} \{ \sin 1 + \sin 2 + (\pi - 3) \sin 3 \} \)

Question. If \( A = \int_{0}^{\pi} \frac{\cos x}{(x + 2)^2} dx \), then \( \int_{0}^{\pi/2} \frac{\sin 2x}{x + 1} dx \) is equal to
(a) \( \frac{1}{2} + \frac{1}{\pi + 2} - A \)
(b) \( \frac{1}{\pi + 2} - A \)
(c) \( 1 + \frac{1}{\pi + 2} - A \)
(d) \( A - \frac{1}{2} - \frac{1}{\pi + 2} \)
Answer: (a) \( \frac{1}{2} + \frac{1}{\pi + 2} - A \)

Question. If \( f(x) = \begin{cases} 0 & , \text{where } x = \frac{n}{n+1}, n = 1, 2, 3 \dots \\ 1 & , \text{else where} \end{cases} \) , then the value of \( \int_{0}^{2} f(x) dx \)
(a) 1
(b) 0
(c) 2
(d) \( \infty \)
Answer: (c) 2

Question. \( \int_{-\pi/2}^{\pi/2} \frac{|x| dx}{8 \cos^2 2x + 1} \) has the value
(a) \( \frac{\pi^2}{6} \)
(b) \( \frac{\pi^2}{12} \)
(c) \( \frac{\pi^2}{24} \)
(d) None of the options
Answer: (b) \( \frac{\pi^2}{12} \)

Question. If \( \int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \), then \( \int_{0}^{\infty} e^{-ax^2} dx \) where a > 0 is
(a) \( \frac{\sqrt{\pi}}{2} \)
(b) \( \frac{\sqrt{\pi}}{2a} \)
(c) \( 2\frac{\sqrt{\pi}}{a} \)
(d) \( \frac{1}{2} \sqrt{\frac{\pi}{a}} \)
Answer: (d) \( \frac{1}{2} \sqrt{\frac{\pi}{a}} \)

Question. The expression \( \frac{\int_{0}^{n} [x] dx}{\int_{0}^{n} \{x\} dx} \) is equal to
(where \( [*] \) and \( \{ * \} \) denotes greatest integer function and fractional part function and \( n \in N \))
(a) \( \frac{1}{n - 1} \)
(b) \( \frac{1}{n} \)
(c) n
(d) n - 1
Answer: (d) n - 1

Question. Let \( A = \int_{0}^{1} \frac{e^t dt}{1 + t} \) then \( \int_{a-1}^{a} \frac{e^{-t}}{t - a - 1} dt \) has the value
(a) \( A e^{-a} \)
(b) \( -A e^{-a} \)
(c) \( -a e^{-a} \)
(d) \( A e^a \)
Answer: (b) \( -A e^{-a} \)

Question. \( \int_{0}^{2n\pi} \left( |\sin x| - \left[ \left| \frac{\sin x}{2} \right| \right] \right) dx \) is equal to
(where \( [ \ \ ] \) denotes the greatest integer function)
(a) 0
(b) 2n
(c) \( 2n\pi \)
(d) 4n
Answer: (d) 4n

Question. A function \( f(x) \) which satisfies, \( f'(\sin^2 x) = \cos^2 x \) for all real \( x \) & \( f(1) = 1 \) is
(a) \( f(x) = x - \frac{x^3}{3} + \frac{1}{3} \)
(b) \( f(x) = x^2 - \frac{x}{2} + \frac{1}{2} \)
(c) a polynomial of degree two
(d) \( f(0) = 1/2 \)
Answer: (c) a polynomial of degree two, (d) \( f(0) = 1/2 \)

Question. If \( I_n = \int_{0}^{1} \frac{dx}{(1 + x^2)^n} \); \( n \in \mathbb{N} \), then which of the following statements hold good ?
(a) \( 2n I_{n+1} = 2^{-n} + (2n - 1) I_n \)
(b) \( I_2 = \frac{\pi}{8} + \frac{1}{4} \)
(c) \( I_2 = \frac{\pi}{8} - \frac{1}{4} \)
(d) \( I_3 = \frac{\pi}{16} - \frac{5}{48} \)
Answer: (a) \( 2n I_{n+1} = 2^{-n} + (2n - 1) I_n \), (b) \( I_2 = \frac{\pi}{8} + \frac{1}{4} \)

Question. If \( f(x) \) is integrable over \( [1, 2] \), then \( \int_{1}^{2} f(x) \,dx \) is equal to
(a) \( \lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} f\left(\frac{r}{n}\right) \)
(b) \( \lim_{n \to \infty} \frac{1}{n} \sum_{r=n+1}^{2n} f\left(\frac{r}{n}\right) \)
(c) \( \lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} f\left(\frac{r+n}{n}\right) \)
(d) \( \lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{2n} f\left(\frac{r}{n}\right) \)
Answer: (b) \( \lim_{n \to \infty} \frac{1}{n} \sum_{r=n+1}^{2n} f\left(\frac{r}{n}\right) \), (c) \( \lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} f\left(\frac{r+n}{n}\right) \)

Question. If \( f(x) = 2^{\{x\}} \), where \( \{x\} \) denotes the fractional part of \( x \). Then which of the following is true ?
(a) f is periodic
(b) \( \int_{0}^{1} 2^{\{x\}} \,dx = \frac{1}{\ln 2} \)
(c) \( \int_{0}^{1} 2^{\{x\}} \,dx = \log_2 e \)
(d) \( \int_{0}^{100} 2^{\{x\}} \,dx = 100 \log_2 e \)
Answer: (a) f is periodic, (b) \( \int_{0}^{1} 2^{\{x\}} \,dx = \frac{1}{\ln 2} \), (c) \( \int_{0}^{1} 2^{\{x\}} \,dx = \log_2 e \), (d) \( \int_{0}^{100} 2^{\{x\}} \,dx = 100 \log_2 e \)

Question. If \( f(x) = \int_{0}^{x} (2\cos^2 3t + 3\sin^2 3t) \,dt \), \( f(x + \pi) \) is equal to
(a) \( f(x) + f(\pi) \)
(b) \( f(x) + 2f\left(\frac{\pi}{2}\right) \)
(c) \( f(x) + 4f\left(\frac{\pi}{4}\right) \)
(d) None of the options
Answer: (a) \( f(x) + f(\pi) \), (b) \( f(x) + 2f\left(\frac{\pi}{2}\right) \)