JEE Mathematics Definite Integration MCQs Set 03

Question. \( f(x) = \text{Minimum } \{ \tan x, \cot x \} \ \forall \ x \in \left( 0, \frac{\pi}{2} \right) \). Then \( \int_{0}^{\pi/3} f(x) dx \) is equal to
(a) \( \ln \left( \frac{\sqrt{3}}{2} \right) \)
(b) \( \ln \left( \sqrt{\frac{3}{2}} \right) \)
(c) \( \ln(\sqrt{2}) \)
(d) \( \ln(\sqrt{3}) \)
Answer: (d) \( \ln(\sqrt{3}) \)

Question. The value of \( \int_{1}^{2} ([x^2] - [x]^2) dx \) is equal to
(where \( [ \ \ ] \) denotes the greatest integer function)
(a) \( 4 + \sqrt{2} - \sqrt{3} \)
(b) \( 4 - \sqrt{2} + \sqrt{3} \)
(c) \( 4 - \sqrt{3} - \sqrt{2} \)
(d) None of the options
Answer: (c) \( 4 - \sqrt{3} - \sqrt{2} \)

Question. If \( f(\pi) = 2 \) and \( \int_{0}^{\pi} (f(x) + f''(x)) \sin x \ dx = 5 \) then f(0) is equal to
(It is given that f(x) is continuous in \( [0, \pi] \))
(a) 7
(b) 3
(c) 5
(d) 1
Answer: (b) 3

Question. If \( u_n = \int_{0}^{\pi/2} x^n \sin x \ dx \), \( n \in N \) then the value of \( u_{10} + 90 u_8 \) is
(a) \( 9 \left( \frac{\pi}{2} \right)^8 \)
(b) \( \left( \frac{\pi}{2} \right)^9 \)
(c) \( 10 \left( \frac{\pi}{2} \right)^9 \)
(d) \( 9 \left( \frac{\pi}{2} \right)^9 \)
Answer: (c) \( 10 \left( \frac{\pi}{2} \right)^9 \)

Question. If \( f(x) = e^{g(x)} \) and \( g(x) = \int_{2}^{x} \frac{t dt}{1 + t^4} \) then \( f '(2) \) has the value equal to
(a) 2/17
(b) 0
(c) 1
(d) Cannot be determined
Answer: (a) 2/17

Question. \( \lim_{n \to \infty} \left( \frac{1}{n^2} \sec^2 \frac{1}{n^2} + \frac{2}{n^2} \sec^2 \frac{4}{n^2} + \dots + \frac{n}{n^2} \sec^2 1 \right) \) equals to
(a) \( \frac{1}{2} \tan 1 \)
(b) \( \tan 1 \)
(c) \( \frac{1}{2} \operatorname{cosec} 1 \)
(d) \( \frac{1}{2} \sec 1 \)
Answer: (a) \( \frac{1}{2} \tan 1 \)

Question. \( \lim_{n \to \infty} \frac{1^p + 2^p + 3^p + \dots + n^p}{n^{p+1}} \) is equal to
(a) \( \frac{1}{p + 1} \)
(b) \( \frac{1}{p - 1} \)
(c) \( \frac{1}{p} - \frac{1}{p - 1} \)
(d) \( \frac{1}{p + 2} \)
Answer: (a) \( \frac{1}{p + 1} \)

Question. Let \( F(x) = f(x) + f \left( \frac{1}{x} \right) \), where \( f(x) = \int_{1}^{x} \frac{\log t}{1 + t} dt \). Then F(e) equals
(a) \( \frac{1}{2} \)
(b) 0
(c) 1
(d) 2
Answer: (a) \( \frac{1}{2} \)

Question. \( \int_{-3\pi/2}^{-\pi/2} [(x + \pi)^3 + \cos^2 (x + 3\pi)] \ dx \) is equal to
(a) \( \left( \frac{\pi^4}{32} \right) + \left( \frac{\pi}{2} \right) \)
(b) \( \left( \frac{\pi}{2} \right) \)
(c) \( \left( \frac{\pi}{4} \right) - 1 \)
(d) \( \frac{\pi^4}{32} \)
Answer: (b) \( \left( \frac{\pi}{2} \right) \)

Question. The solution for x of the equation \( \int_{\sqrt{2}}^{x} \frac{dt}{t \sqrt{t^2 - 1}} = \frac{\pi}{2} \) is
(a) \( -\sqrt{2} \)
(b) \( \pi \)
(c) \( \frac{\sqrt{3}}{2} \)
(d) \( 2\sqrt{2} \)
Answer: (a) \( -\sqrt{2} \)

Question. \( \int_{0}^{\pi} x f(\sin x) \ dx \) is equal to
(a) \( \pi \int_{0}^{\pi} f(\sin x) \ dx \)
(b) \( \frac{\pi}{2} \int_{0}^{\pi/2} f(\sin x) \ dx \)
(c) \( \pi \int_{0}^{\pi/2} f(\cos x) \ dx \)
(d) \( \pi \int_{0}^{\pi} f(\cos x) \ dx \)
Answer: (c) \( \pi \int_{0}^{\pi/2} f(\cos x) \ dx \)

Question. The value of \( \int_{1}^{a} [x] f'(x) dx \), a > 1, where [x] denotes the greatest integer not exceeding x, is
(a) \( [a] f(a) - \{ f(1) + f(2) + \dots + f([a]) \} \)
(b) \( [a] f([a]) - \{ f(1) + f(2) + \dots + f(a) \} \)
(c) \( a f([a]) - \{ f(1) + f(2) + \dots + f(a) \} \)
(d) \( af(a) - \{ f(1) + f(2) + \dots + f([a]) \} \)
Answer: (a) \( [a] f(a) - \{ f(1) + f(2) + \dots + f([a]) \} \)

Question. Let \( f : R \to R \) be a differentiable function having \( f(2) = 6, f '(2) = \left( \frac{1}{48} \right) \). Then \( \lim_{x \to 2} \int_{6}^{f(x)} \frac{4t^3}{x - 2} dt \) equals
(a) 18
(b) 12
(c) 36
(d) 24
Answer: (a) 18

Question. If \( I_1 = \int_{0}^{1} 2^{x^2} dx \), \( I_2 = \int_{0}^{1} 2^{x^3} dx \), \( I_3 = \int_{1}^{2} 2^{x^2} dx \) and \( I_4 = \int_{1}^{2} 2^{x^3} dx \), then
(a) \( I_3 > I_4 \)
(b) \( I_3 = I_4 \)
(c) \( I_1 > I_2 \)
(d) \( I_2 > I_1 \)
Answer: (c) \( I_1 > I_2 \)

Question. The value of \( \int_{0}^{\pi/2} \frac{(\sin x + \cos x)^2}{\sqrt{1 + \sin 2x}} dx \) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (c) 2

Question. If \( f(x) = \frac{e^x}{1 + e^x} \), \( I_1 = \int_{f(-a)}^{f(a)} x g\{ x(1 - x) \} dx \) and \( I_2 = \int_{f(-a)}^{f(a)} g\{ x(1 - x) \} dx \), then the value of \( \frac{I_2}{I_1} \) is
(a) 2
(b) -3
(c) -1
(d) 1
Answer: (a) 2

Question. If \( f(y) = e^y, g(y) = y; y > 0 \) and \( F(t) = \int_{0}^{t} f(t - y) g(y) dy \), then
(a) \( F(t) = 1 - e^{-t}(1 + t) \)
(b) \( F(t) = e^t - (1 + t) \)
(c) \( F(t) = t e^t \)
(d) \( F(t) = t e^{-t} \)
Answer: (b) \( F(t) = e^t - (1 + t) \)

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

Question. The value of \( \lim_{x \to 0} \frac{\int_{0}^{x^2} \sec^2 t \ dt}{x \sin x} \) is
(a) 3
(b) 2
(c) 1
(d) -1
Answer: (c) 1

Question. The value of the integral \( l = \int_{0}^{1} x(1 - x)^n dx \) is
(a) \( \frac{1}{n + 1} \)
(b) \( \frac{1}{n + 2} \)
(c) \( \frac{1}{n + 1} - \frac{1}{n + 2} \)
(d) \( \frac{1}{n + 1} + \frac{1}{n + 2} \)
Answer: (c) \( \frac{1}{n + 1} - \frac{1}{n + 2} \)

Question. Let \( \frac{d}{dx} F(x) = \left( \frac{e^{\sin x}}{x} \right), x > 0 \). If \( \int_{1}^{4} \frac{3}{x} e^{\sin x^3} dx = F(k) - F(1) \), then one of the possible values of k, is
(a) 15
(b) 16
(c) 63
(d) 64
Answer: (d) 64

Question. Let f(x) be a function satisfying \( f '(x) = f(x) \) with f(0) = 1 and g(x) be a function that satisfies \( f(x) + g(x) = x^2 \). Then the value of the integral \( \int_{0}^{1} f(x) g(x) dx \), is
(a) \( e - \frac{e^2}{2} - \frac{5}{2} \)
(b) \( e + \frac{e^2}{2} - \frac{3}{2} \)
(c) \( e - \frac{e^2}{2} - \frac{3}{2} \)
(d) \( e + \frac{e^2}{2} + \frac{5}{2} \)
Answer: (c) \( e - \frac{e^2}{2} - \frac{3}{2} \)

Question. \( \int_{0}^{\pi/2} \frac{dx}{1 + \tan^3 x} \) is equal to
(a) 0
(b) \( \pi/2 \)
(c) \( \pi/3 \)
(d) \( \pi/4 \)
Answer: (d) \( \pi/4 \)

Question. \( \int_{\sin x}^{1} t^2 f(t) dt = 1 - \sin x \ \forall \ x \in (0, \pi/2) \), then \( f \left( \frac{1}{\sqrt{3}} \right) \) is
(a) 3
(b) \( \sqrt{3} \)
(c) 1/3
(d) None of the options
Answer: (a) 3

Question. If \( I_n = \int_{0}^{\pi/4} \tan^n x \ dx \), then \( \frac{1}{I_2 + I_4}, \frac{1}{I_3 + I_5}, \frac{1}{I_4 + I_6} \) is
(a) A.P.
(b) G.P.
(c) H.P.
(d) None of the options
Answer: (a) A.P.

Question. \( \lim_{x \to 0} \frac{\int_{0}^{x^2} \cos t^2 \ dt}{x \sin x} \) is equal to
(a) -1
(b) 1
(c) 2
(d) -2
Answer: (b) 1

Question. \( \int_{0}^{\pi/4} \sin (x - [x]) d(x - [x]) \) 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 [x] denotes the greatest integer less than or equal to x, then the value of \( \int_{1}^{5} [|x - 3|] dx \) is
(a) 1
(b) 2
(c) 4
(d) 8
Answer: (b) 2

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

Question. If \( \frac{C_0}{1} + \frac{C_1}{2} + \frac{C_2}{3} = 0 \), where \( C_0, C_1, C_2 \) are all real, the equation \( C_2 x^2 + C_1 x + C_0 = 0 \) has
(a) atleast one root in (0, 1)
(b) one root in (1, 2) & other in (3, 4)
(c) one root in (-1, 1) & the other in (-5, -2)
(d) both roots are imaginary
Answer: (a) atleast one root in (0, 1)

Question. If f(x) satisfies the requirements of Rolle's Theorem in [1, 2] and f '(x) is continuous in [1, 2], then \( \int_{1}^{2} f '(x) dx \) is equal to
(a) 0
(b) 1
(c) 3
(d) -1
Answer: (a) 0

Question. \( \int_{1}^{2} (x - \log_2 a) dx = 2 \log_2 \left( \frac{2}{a} \right) \), if
(a) a > 0
(b) a > 2
(c) a = 4
(d) a = 8
Answer: (a) a > 0

Question. \( \int_{-1}^{1} \frac{x^4}{1 + e^{x^7}} dx \) is
(a) \( \frac{1}{2} \)
(b) 0
(c) \( \frac{1}{5} \)
(d) None of the options
Answer: (c) \( \frac{1}{5} \)

Question. \( \frac{1}{c} \int_{ac}^{bc} f \left( \frac{x}{c} \right) dx = \)
(a) \( \frac{1}{c} \int_{a}^{b} f(x) dx \)
(b) \( \int_{a}^{b} f(x) dx \)
(c) \( c \int_{a}^{b} f(x) dx \)
(d) \( \int_{ac^2}^{bc^2} f(x) dx \)
Answer: (b) \( \int_{a}^{b} f(x) dx \)

Question. If \( \int_{\ln 2}^{x} \frac{dx}{\sqrt{e^x - 1}} = \frac{\pi}{6} \), then x =
(a) 4
(b) \( \ln 8 \)
(c) \( \ln 4 \)
(d) None of the options
Answer: (c) \( \ln 4 \)

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

Question. \( \int_{0}^{\infty} \frac{x}{(1 + x)(1 + x^2)} \,dx \)
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{2} \)
(c) is same as \( \int_{0}^{\infty} \frac{dx}{(1 + x)(1 + x^2)} \)
(d) cannot be evaluated
Answer: (a) \( \frac{\pi}{4} \), (c) is same as \( \int_{0}^{\infty} \frac{dx}{(1 + x)(1 + x^2)} \)

Question. The value of integral \( \int_{a}^{b} \frac{|x|}{x} \,dx \), \( a < b \) is
(a) \( b - a \) if \( a > 0 \)
(b) \( a - b \) if \( b < 0 \)
(c) \( b + a \) if \( a < 0 < b \)
(d) \( |b| - |a| \)
Answer: (a) \( b - a \) if \( a > 0 \), (b) \( a - b \) if \( b < 0 \), (c) \( b + a \) if \( a < 0 < b \), (d) \( |b| - |a| \)

Question. If \( f(x) = \int_{0}^{x} (\cos^4 t + \sin^4 t) \,dt \), \( f(x + \pi) \) will be equal to
(a) \( f(x) + f(\pi) \)
(b) \( f(x) + 2(\pi) \)
(c) \( f(x) + f\left(\frac{\pi}{2}\right) \)
(d) \( f(x) + 2f\left(\frac{\pi}{2}\right) \)
Answer: (a) \( f(x) + f(\pi) \), (d) \( f(x) + 2f\left(\frac{\pi}{2}\right) \)

Question. The value of \( \int_{0}^{1} \frac{2x^2 + 3x + 3}{(x + 1)(x^2 + 2x + 2)} \,dx \) is
(a) \( \frac{\pi}{4} + 2\ln 2 - \tan^{-1} 2 \)
(b) \( \frac{\pi}{4} + 2\ln 2 - \tan^{-1} 3 \)
(c) \( 2\ln 2 - \cot^{-1} 3 \)
(d) \( -\frac{\pi}{4} + \ln 4 + \cot^{-1} 2 \)
Answer: (a) \( \frac{\pi}{4} + 2\ln 2 - \tan^{-1} 2 \), (d) \( -\frac{\pi}{4} + \ln 4 + \cot^{-1} 2 \)