CBSE Questions for Class 12 Commerce Applied Mathematics Definite Integrals Quiz 11 - MCQExams.com

$$\displaystyle \int _{ 0 }^{ \pi /2 }{ \dfrac { dx }{ 2+\cos { x }  } = }$$
  • $$\dfrac { 2}{ \sqrt { 3 } } \tan ^{ -1 }{ \left( \dfrac { 1 }{ \sqrt { 3 } } \right) }$$
  • $$\sqrt { 3 } \tan ^{ -1 }{ \left( \sqrt { 3 } \right) }$$
  • $$\dfrac { 1 }{ \sqrt { 2 } } \tan ^{ -1 }{ \left( \dfrac { 1 }{ \sqrt { 3 } } \right) }$$
  • $$2\sqrt { 3 } \tan ^{ -1 }{ \left( \sqrt { 3 } \right) }$$
Evaluate: $$\displaystyle \overset{\pi/3}{\underset{\pi/6}{\int}} \dfrac{dx}{1 + \sqrt{\tan x}}$$
  • $$\frac{\pi}{3}$$
  • $$\frac{\pi}{6}$$
  • $$\frac{2\pi}{3}$$
  • $$\frac{\pi}{12}$$
$$\displaystyle \int^{\pi/2}_{-\pi/2}\sqrt {\cos x-\cos^{3}x}dx=$$
  • $$1$$
  • $$4/3$$
  • $$-1/3$$
  • $$0$$
$${ I }_{ n }=\int _{ 0 }^{ 1 }{ { \left( 1-{ x }^{ 50 } \right)  }^{ n }dx }$$, then $$\dfrac { { I }_{ 100 } }{ { I }_{ 101 } }$$
  • $$\cfrac{{I}_{100}}{{I}_{101}} = \cfrac{5001}{5500}$$.
  • $$\cfrac{{I}_{100}}{{I}_{101}} = \cfrac{5051}{5050}$$.
  • $$\cfrac{{I}_{100}}{{I}_{101}} = \cfrac{5050}{5550}$$.
  • $$\cfrac{{I}_{100}}{{I}_{101}} = \cfrac{5510}{5055}$$.
$$\int_\limits{0}^1 \dfrac{x^3}{\sqrt{1-x^2}}dx$$
  • $$2\dfrac{\pi}{3}$$
  • $$\dfrac{2}{3}$$
  • $$\dfrac{3}{2}$$
  • None of these
$$\int_{0}^{1}\dfrac{\sin t}{1+t}dt=\alpha$$, then the value of $$\int_{4\pi-2}^{4\pi}\dfrac{\sin (t/2)}{4\pi+2-t}dt=$$
  • $$\alpha$$
  • $$-\alpha$$
  • $$2\alpha$$
  • $$4\pi-2\alpha$$
If $$(-1, 2)$$ and $$(2, 4)$$ are two points on the curve $$y=f(x)$$ and if $$g(x)$$ is the gradient of the curve at point (x, y), then the value of the integral $$\displaystyle\int^{2}_{-1}g(x)dx$$, is?
  • $$2$$
  • $$-2$$
  • $$0$$
  • $$1$$
What is the value of $$\int_{0}^{\pi}\dfrac {dx}{5-4\cos x}$$?
  • $$\dfrac {\pi \log 2}{32}$$
  • $$dfrac {4\pi}{7}$$
  • $$dfrac {\pi}{3}$$
  • $$None\ of\ these$$
Solve $$\displaystyle\int_{0}^{\infty}\dfrac {x \tan^{-1}x}{(1+x^{2})^{2}}dx$$
  • $$\pi/2$$
  • $$\pi/6$$
  • $$\pi/4$$
  • $$\pi/8$$
The value of $$\int_{-1}^{1}\dfrac {dx}{(2-x)\sqrt {1-x^{2}}}$$ is
  • $$0$$
  • $$\dfrac {\pi}{\sqrt {3}}$$
  • $$\dfrac {2\pi}{\sqrt {3}}$$
  • $$cannot\ be\ evaluated$$
If $$\displaystyle \int_{1}^{x}\dfrac{dt}{|t|\sqrt{t^{2}-1}}=\dfrac{\pi}{6}$$, then $$x$$ can be equal to :
  • $$\dfrac{2}{\sqrt{3}}$$
  • $$\sqrt{3}$$
  • $$2$$
  • $$\dfrac{4}{\sqrt{3}}$$
Solve $$\displaystyle\int_{0}^{\pi}\dfrac {1}{3+2\sin x+\cos x}dx$$
  • $$\dfrac{5\pi}{4}$$
  • $$\dfrac{\pi}{4}$$
  • $$-\dfrac{\pi}{4}$$
  • None of these
The integral $$\int _{ 0 }^{ \pi  }{ \sqrt { 1+4\sin { ^{ 2 }\frac { x }{ 2 } -4 } \sin { \frac { x }{ 2 }  }  } dx }$$ is equal to 
  • $$\pi-4$$
  • $$\frac{2\pi}{3}-4-4\sqrt{3}$$
  • $$4sqrt{3}-4$$
  • $$4sqrt{3}-4-\frac{\pi}{3}$$
$$\int _{ 1/2 }^{ 2 }{ \dfrac { 1 }{ x } \csc { ^{ 101 }\left( x-\dfrac { 1 }{ x }  \right) dx } \\  } $$ is equal to 
  • $$1/4$$
  • $$1$$
  • $$0$$
  • $$\dfrac{101}{2}$$
The value of $$\int _{ 1/e }^{ \tan { x }  }{ \dfrac { t }{ 1+{ t }^{ 2 } } dt+ } \int _{ 1/e }^{ \cot { x }  }{ \dfrac { t }{ t\left( 1+{ t }^{ 2 } \right)  } dt } $$, where $$x\in \left( \pi /6,\ \pi /3 \right) $$, is equal to :
  • $$0$$
  • $$2$$
  • $$1$$
  • $$cannot\ be\ determined$$
If $$G(x) = \begin{vmatrix}f(x)f(-x) & 0 & x^{4}\\ 3 & f(x) - f(-x) & \cos x\\ x^{4} & 2x & f(x)f(-x)\end{vmatrix}$$, then $$\int_{-2}^{2} x^{4}G(x)dx$$ is equal to
  • $$-1$$
  • $$0$$
  • $$2$$
  • $$1$$
The value of $$\int\limits_0^\pi {\frac{{{e^{\cos x}}}}{{{e^{\cos x}} + {e^{ - \cos x}}}}dx} $$
  • $$\pi $$
  • $$\frac{\pi }{2}$$
  • $$\frac{\pi }{4}$$
  • $$\frac{\pi }{5}$$
The value of $$\displaystyle \int\limits_0^1 {\dfrac{{x{{\tan }^{ - 1}}x}}{{{{\left( {1 + {x^2}} \right)}^{3/2}}}}} dx$$ is 
  • $$\dfrac{{4 + \pi }}{{4\sqrt 2 }}$$
  • $$\dfrac{{4 - \pi }}{{4\sqrt 2 }}$$
  • $$\dfrac{\pi }{2}$$
  • $$ - \dfrac{\pi }{2}$$
$$\int _{ 0 }^{ 1 }{ \frac { x }{ 1+\sqrt { x }  } dx= }$$
  • $$\frac {5}{3}-\log {4}$$
  • $$\frac {5}{3}+\log {4}$$
  • $$\frac {5}{3}\log {4}$$
  • $$\frac {3}{5}-\log {4}$$
Solve $$\int\limits_0^\pi  {\log {{\sin }^2}x} dx$$
  • $$2\pi \log \left( {1/2} \right)$$
  • $$\pi \log 2$$
  • $$\pi /2\log \left( {1/2} \right)$$
  • None of these
The value of the integral $$\int _{ \dfrac { 1 }{ 3 }  }^{ 1 }{ \dfrac { \left( x-{ x }^{ 3 } \right) ^{ \dfrac { 1 }{ 3 }  } }{ { x }^{ 4 } }  }dx $$ is
  • 6
  • 0
  • 3
  • 4
$$\overset { -5 }{ \underset { -4 }{ \int }  } e^(x + 5)^2 dx + 3  \overset { 2/3}{ \underset { 1/3 }{ \int }  } e^9(9(x-2/3)^2$$ dx is equal toi 
  • $$e^5$$
  • $$e^4$$
  • $$3e^2$$
  • 0
If $$I=\displaystyle\int^1_0\cos\left(2\cot^{-1}\sqrt{\left(\dfrac{1-x}{1+x}\right)}\right)dx$$ then?
  • $$I > \dfrac{1}{2}$$
  • $$I=-\dfrac{1}{2}$$
  • $$0 < I < \dfrac{1}{2}$$
  • None of these
If $$\overset { 1 }{ \underset { 0 }{\int  }  } 2^{x^2}dx,I_2=\overset { 1 }{ \underset { 0 }{\int  }  }2^{x^3}dx,I_3 =\overset { 2 }{ \underset { 1 }{\int  }  }2^{x^2}dx, $$ and $$I_4=\overset { 1 }{ \underset { 0 }{\int  }  }2^{x^3}dx $$ then-
  • $$I_2>I_1$$
  • $$I_1>I_2$$
  • $$I_3=I_4$$
  • $$I_3>I_4$$
$$\overset { 1 }{ \underset { -1 }{ \int }  } \dfrac{x^3+|x|+1}{x^2+2|x|+1}$$dx is equal to
  • $$In 3$$
  • $$2In 3$$
  • $$\dfrac{1}{3}In 3$$
  • none of these
The value of  $$\int\limits_1^2 {\frac{1}{{{x^2}}}{e^{ - 1/x}}} dx$$ is 
  • $$\frac{1}{{\sqrt e }} + \frac{1}{e}$$
  • $$\frac{1}{e} - \frac{1}{{\sqrt e }}$$
  • $$\frac{1}{{\sqrt e }} - \frac{1}{e}$$
  • 0
The value of $$\int_{-1}^{3}[tan^{-1}(\frac{x}{x^{2}+1})+tan^{-1}(\frac{x^{2}+1}{x})]dx$$ 
  • 2$$\pi $$
  • $$\pi $$
  • $$\frac{\pi }{2}$$
  • $$\frac{\pi }{4}$$
The value of the integral $$\underset{0}{\overset{\pi / 4}{\int}} \dfrac{\sin \, x + \cos \, x}{3 + \sin \, 2x} dx$$, is 
  • $$\log 2$$
  • $$\log 3$$
  • $$\dfrac{1}{4} \log \, 3$$
  • $$\dfrac{1}{8} \log \, 3$$
$$\int^{\pi/4}_{0}\dfrac {\sin^{2}x\cos^{2}x}{(\sin^{3}x+\cos^{3}x)^{2}}dx$$ is
  • $$1/3$$
  • $$1/2$$
  • $$1/6$$
  • $$1/4$$
If $$I = \int _ { 0 } ^ { 1 } \frac { \tan x } { \sqrt { x } } d x$$ then ?
  • $$I < \frac { 2 } { 3 }$$
  • $$I > \frac { 2 } { 3 }$$
  • $$I < \frac { 5 } { 9 }$$
  • $$I < \frac { 1 } { 3 }$$
$$\int _ { 0 } ^ { \pi ^ { 2 } / 4 } \sin \sqrt { x } d x$$ is
  • 0
  • 1
  • 2
  • 4
$$\displaystyle \int_{0}^{\pi/4}{\dfrac{x.\sin x}{\cos^{3}x}dx}$$ equals to :
  • $$\dfrac{\pi}{4}+\dfrac{1}{2}$$
  • $$\dfrac{\pi}{4}-\dfrac{1}{2}$$
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{4}+1$$
The value of $$\int_{a}^{b}{(x-a)^{3}(b-x)^{4}dx}$$ is
  • $$\dfrac{(b-a)^{4}}{6^{4}}$$
  • $$\dfrac{(b-a)^{8}}{280}$$
  • $$\dfrac{(b-a)^{7}}{7^{3}}$$
  • $$none\ of\ these$$
$$\displaystyle\int^1_0\tan^{-1}\left[\dfrac{2x-1}{1+x-x^2}\right]dx=?$$
  • $$0$$
  • $$1/2$$
  • $$1$$
  • $$\pi/6$$
Consider $$f ( x ) = \int _ { 0 } ^ { \pi } \frac { \ln ( 1 + x \cos \theta ) } { \cos \theta } d \theta$$
Range of $$f ( x )$$ is
  • $$( 0 , \pi )$$
  • $$\left( 0 , \pi ^ { 2 } \right)$$
  • $$\left( \frac { - \pi } { 2 } , \frac { \pi } { 2 } \right)$$
  • $$\left( \frac { - \pi ^ { 2 } } { 2 } , \frac { \pi ^ { 2 } } { 2 } \right)$$
$$\displaystyle\int _{ -1 }^{ 1 }{ x\ell n\left( 1+{ e }^{ x } \right) dx } =$$
  • $$0$$
  • $$\ell n(1+e)$$
  • $$\ell n(1+e)-1$$
  • $$1/3$$
If $$\displaystyle \int _{0}^{1}\cot^{-1}(1+x^{2}-x)dx=k\left(\dfrac {\pi}{4}-\log_{e}\sqrt {2}\right)$$, then the value of $$k$$ is equal to
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$2$$
$$\int _{ \pi /4 }^{ 3\pi /4 }{ \dfrac { dx }{ 1+\cos { x }  }  }$$ is equal to 
  • $$-2$$
  • $$2$$
  • $$4$$
  • $$-1$$
The value of $$\displaystyle \int _{0}^{\pi/2} \log{\sin{x}}dx$$ is 
  • $$-\pi \log{2}$$
  • $$\dfrac{-\pi}{2} \log{2}$$
  • $$\pi \log{2}$$
  • 0
The value of $$\displaystyle \int _{-1}^{1} \log{\left(\dfrac{2-x}{2+x}\right)}\sin^{2}{x}dx$$
  • $$1$$
  • $$-1$$
  • $$2$$
  • $$0$$
The value of the definite intergral $$\displaystyle \int_{19}^{37}($${x}$$^2$$ +$$3\sin (2\pi x))dx$$, where {.} denotes the fractional part function
  • $$0$$
  • $$6$$
  • $$9$$
  • can not determine
The value of the integral $$\displaystyle \int _{ { e }^{ -1 } }^{ { e }^{ 2 } }{ \left| \frac { \log _{ e }{ x }  }{ x }  \right| dx } $$ is
  • $$\dfrac {3}{2}$$
  • $$\dfrac {5}{2}$$
  • $$3$$
  • $$5$$
$$\displaystyle\int^{2+\sqrt{3}}_{2-\sqrt{3}}\dfrac{xdx}{(1+x)(1+x^2)}=?$$
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{12}$$
  • $$\dfrac{\pi}{24}$$
Let $$I=\overset{1}{\underset{0}{\int}}\dfrac{\sin x}{\sqrt x}dx$$ and $$J=\overset{1}{\underset{0}{\int}}\dfrac{\cos x}{\sqrt x}dx$$. Then which one of the following is true?
  • $$1>\dfrac{2}{3}$$ and $$J>2$$
  • $$1<\dfrac{2}{3}$$ and $$J<2$$
  • $$1<\dfrac{2}{3}$$ and $$J>2$$
  • $$1>\dfrac{2}{3}$$ and $$J<2$$
The area of the region bounded by the lines $$x = 1, x = 2$$, and the curves $$x(y - e^x) = \sin x$$ and $$2xy = 2 \sin x + x^3$$ is 
  • $$e^2 - e - \dfrac{1}{6}$$
  • $$e^2 - e - \dfrac{7}{6}$$
  • $$e^2 - e + \dfrac{1}{6}$$
  • $$e^2 - e + \dfrac{7}{6}$$
The value of the integral $$\displaystyle \int _ { 0 } ^ { 1 } \dfrac { d x } { x ^ { 2 } + 2 x \cos \alpha + 1 }$$ , where $$0 < \alpha < \dfrac { \pi } { 2 } ,$$ is equal to
  • $$\sin{\alpha}$$
  • $$\alpha \sin {\alpha}$$
  • $$\dfrac { \alpha } { 2 \sin{\alpha} }$$
  • $$\dfrac { \alpha } { 2 } \sin {\alpha}$$
$$\displaystyle\int^{\lambda}_0\dfrac{y}{\sqrt{y+\lambda}}dy=?$$
  • $$\dfrac{2}{3}(2-\sqrt{2})\lambda \sqrt{\lambda}$$
  • $$\dfrac{2}{3}(2+\sqrt{2})\lambda\sqrt{\lambda}$$
  • $$\dfrac{1}{3}(2-\sqrt{2})\lambda \sqrt{\lambda}$$
  • $$\dfrac{1}{3}(2+\sqrt{2})\lambda \sqrt{\lambda}$$
If $$\int _ { \ln 2 } ^ { x } \frac { d x } { \sqrt { e ^ { x } - 1 } } = \frac { \pi } { 6 } ,$$ then $$x$$ is equal to
  • $$\ln 8$$
  • $$\ln 2$$
  • $$\ln 4$$
  • $$4$$
$$\displaystyle\int^{2\pi}_0[\sin x]dx$$.
  • 0
  • $$-\pi$$
  • $$2\pi$$
  • $$-2\pi$$
$$\displaystyle \int _ { 0 } ^ { \pi / 4 } \frac { x \cdot \sin x } { \cos ^ { 3 } x } d x$$ equals to :
  • $$\displaystyle \frac { \pi } { 4 } + \frac { 1 } { 2 }$$
  • $$\displaystyle \frac { \pi } { 4 } - \frac { 1 } { 2 }$$
  • $$\dfrac { \pi } { 4 }$$
  • $$\dfrac { \pi } { 4 } + 1$$
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