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CBSE Questions for Class 12 Commerce Applied Mathematics Definite Integrals Quiz 9 - MCQExams.com

Let I1=2111+x2dx and I2=211xdx. Then
  • I1>I2
  • I2>I1
  • I1=I2
  • I1>2I2
The value (s) of  \displaystyle \int_{0}^{1}\frac{x^{4}\left ( 1-x \right )^{4}}{1+x^{2}} dx is (are)
  • \displaystyle \frac{22}{7}-\pi
  • \displaystyle \frac{2}{105}
  • 0
  • \displaystyle \frac{71}{15}-\frac{3\pi}{2}
If \displaystyle I= \int_{0}^{1}\frac{x dx}{8+x^{3}} then the smallest interval in which I lies is
  • \displaystyle \left ( 0,\frac{1}{8} \right )
  • \displaystyle \left ( 0,\frac{1}{9} \right )
  • \displaystyle \left ( 0,\frac{1}{10} \right )
  • \displaystyle \left ( 0,\frac{1}{7} \right )
If \displaystyle I_{n} =\int_{0}^{\frac{\pi }{4}}\tan ^{n}xdx 

then \displaystyle \frac{1}{I_{2}+I_{4}},\frac{1}{I_{3}+I_{5}},\frac{1}{I_{4}+I_{6}} are in?
  • A.P
  • H.P
  • G.P
  • None of these
If \displaystyle I_{t}=\int_{0}^{\dfrac{\pi }{2}}\frac{\sin^{2}tx}{\sin^{2}x}dx then ,I_{1},I_{2},I_{3} are in
  • A.P.
  • H.P.
  • G.P.
  • None of these
The value of \displaystyle \int_{0}^{\pi /2}\sin \theta \log \left ( \sin \theta \right )\:d\theta   equals
  • \displaystyle \log_{e}\left ( \frac{1}{e} \right )
  • \displaystyle \log _{2}e
  • \displaystyle \log_{e}{2}-1
  • \displaystyle \log_{e}\left ( \frac{e}{2} \right )
If f(x) =\displaystyle  \underset{1}{\overset{x}{\int}} \dfrac{\tan^{-1} t}{t} dt \, \, \forall \in R, then the value of f(e^2) - f \left (\dfrac{1}{e^2}\right) is 
  • 0
  • \dfrac{\pi}{2}
  • \pi
  • 2 \pi
The value(s) of \int _{ 0 }^{ 1 }{ \cfrac { { x }^{ 4 }{ \left( 1-x \right)  }^{ 4 } }{ 1+{ x }^{ 2 } }  } dx is (are)
  • \cfrac{22}{7}-\pi
  • \cfrac{2}{105}
  • 0
  • \cfrac{71}{15}-\cfrac{3\pi}{2}
If \displaystyle 2f(x)+f(-x)=\frac{1}{x}\sin{\left(x-\frac{1}{x}\right)}, then the value of \displaystyle\int_{\frac{1}{e}}^{e}{f(x)dx}, is
  • 1
  • 0
  • e
  • -1
If \displaystyle I= \int_{1/\pi }^{\pi }\frac{1}{x}\cdot \sin \left ( x-\frac{1}{x} \right )dx then I is equal to
  • 0
  • \displaystyle \pi
  • \displaystyle \pi -\frac{1}{\pi }
  • \displaystyle \pi +\frac{1}{\pi }
If x satisfies the equation \displaystyle\left(\int_0^1{\frac{dt}{t^2+2t\cos{\alpha}+1}}\right)x^2-\left(\int_{-3}^3{\frac{t^2\sin{2t}}{t^2+1}dt}\right)x-2=0 
for (0<\alpha<\pi)
then the value of x is?
  • \displaystyle\pm\sqrt{\frac{\alpha}{2\sin{\alpha}}}
  • \displaystyle\pm\sqrt{\frac{2\sin{\alpha}}{\alpha}}
  • \displaystyle\pm\sqrt{\frac{\alpha}{\sin{\alpha}}}
  • \displaystyle\pm2\sqrt{\frac{\sin{\alpha}}{\alpha}}
The tangent to the graph of the function \displaystyle y = f(x) at the point with abscissa x = a forms with the x-axis an angle of \displaystyle \pi/3 and at the point with abscissa x = b at an angle of \displaystyle \pi/4, then the value of the integral,
\displaystyle \int_{a}^{b} f'(x).f''(x)dx is equal to
  • 1
  • 0
  • \displaystyle -\sqrt 3
  • -1
Let \displaystyle F\left ( x \right )=f\left ( x \right )+f\left ( \frac{1}{x} \right ) where \displaystyle f\left ( x \right )=\int_{1}^{x}\frac{\log t}{1+t}dt 
Then F(e) is equal to?
  • 1
  • 2
  • 1/2
  • 0
The value of \displaystyle \int _{ 0 }^{ \pi /2 }{ \frac { d\theta  }{ 5+3\cos { \theta  }  }  } is?
  • \displaystyle \tan ^{ -1 }{ \frac { 1 }{ 2 }  }
  • \displaystyle \tan ^{ -1 }{ \frac { 1 }{ 3 }  }
  • \displaystyle \frac{1}{2}\tan ^{ -1 }{ \frac { 1 }{ 2 }  }
  • \displaystyle \frac{1}{3}\tan ^{ -1 }{ \frac { 1 }{ 3 }  }
Evaluate : \displaystyle \underset{0}{\overset{\infty}{\int}} \dfrac{dx}{(1 + x^2)^4}
  • \dfrac{\pi}{32}
  • \dfrac{3 \pi}{32}
  • \dfrac{5 \pi}{32}
  • \dfrac{7 \pi}{32}
Let f:R\rightarrow R^{+} and I_{I}=\int^{k}_{1-k}\,xf(x(1-x))\,dx,I_2=\int^{k}_{1-k}f(x(1-x))\,dx where 2k-1>0. Then \dfrac{I_I}{I_2} is 
  • 2
  • k
  • 1/2
  • 1
The value of \int _{ 0 }^{ \infty  }{ \cfrac { \log { x }  }{ { a }^{ 2 }+{ x }^{ 2 } }  }
  • \cfrac{2\pi\log{a}}{a}
  • \cfrac{\pi\log{a}}{2a}
  • \pi \log{a}
  • 0
\displaystyle \int_{-1}^{1} \cot^{-1} \left(\dfrac{x+x^{3}}{1+x^{4}}\right)dx is equal to 
  • 2\pi
  • \dfrac{\pi}{2}
  • 0
  • \pi
\displaystyle \int_{0}^{\pi /2}\sin x\log \left ( \sin x \right )dx=
  • \displaystyle \log _{e}e
  • \displaystyle \log _{e}2
  • \displaystyle \log _{e}\left ( e/2 \right )
  • \displaystyle \log _{e}\left ( 2/e \right )
\displaystyle \int_{0}^{\infty}\frac{1}{1+x^{n}}dx,\:\forall\:n\:> 1 is equal to?
  • \displaystyle2 \int_{0}^{\infty}\frac{1}{1+x}dx
  • \displaystyle \int_{-\infty}^{\infty}\frac{1}{1+x^{n}}dx
  • \displaystyle \int_{1}^{\infty}\frac{dx}{(x^{n}-1)^{1/n}}
  • \displaystyle \int_{0}^{1}\frac{1}{(1-x^{n})^{1/n}}
If \displaystyle I_1=\int_{0}^{\pi /2}\frac{x}{\sin x}dx and \displaystyle I_2=\int_{0}^{\pi /2}\frac{\tan ^{-1}x}{x}dx, then \displaystyle \frac{I_{1}}{I_{2}}=  

  • \displaystyle \frac{1}{2}
  • 1
  • 2
  • \displaystyle \frac{\pi }{2}
\displaystyle If \int_{-1}^{1}\frac{g\left ( x \right )}{1+t^{2}}dt= f\left ( x \right ) , where,  g\left ( x \right )= \sin x , then {f}'\left ( \frac{\pi }{3} \right ) equals

  • \displaystyle\frac{\pi }{4}
  • does not exist
  • \displaystyle \frac{\pi \sqrt{3}}{4}
  • None of these
The value of \displaystyle \int_{1}^{1/e}f(x)dx+\int_{1}^{e}f(x)dx where f(x) is given as \displaystyle \frac{log\:x}{1+x} equals 
  • 0
  • \displaystyle -\frac{1}{2}
  • \displaystyle \frac{1}{2}
  • 1
The value of \displaystyle \int_{1/e}^{\tan x}\displaystyle \frac{t}{1+t^{2}}\, dt\, +\, \displaystyle \int_{1/e}^{\cot x}\displaystyle \frac{dt}{t\left ( 1+t^{2} \right )} is
  • 1/2
  • 1
  • \pi /4
  • none of these
Evaluate : \displaystyle \int_{-\frac{1}{\sqrt2}}^{\frac{1}{\sqrt2}}\frac{x^{8}}{1-x^{4}}\times \left [ \sin ^{-1}\left ( 1-2x^{2} \right ) +\cos ^{-1}\left ( 2x\sqrt{1-x^{2}} \right )\right ]dx
  • \displaystyle \pi \left [ \frac{1}{2}\log \frac{\sqrt{2}+1}{\sqrt{2}-1}+\tan ^{-1} \frac{1}{\sqrt{2}}-\frac{21}{10\sqrt{2}}\right ]
  • \displaystyle \pi \left [ \frac{1}{2}\log \frac{\sqrt{2}+1}{\sqrt{2}-1}+\tan ^{-1} \frac{1}{\sqrt{2}}+\frac{21}{10\sqrt{2}}\right ]
  • \displaystyle \pi \left [ \frac{1}{2}\log \frac{\sqrt{2}-1}{\sqrt{2}-1}+\tan ^{-1} \frac{1}{\sqrt{2}}-\frac{21}{10\sqrt{2}}\right ]
  • \displaystyle \pi \left [ \frac{1}{2}\log \frac{\sqrt{2}-1}{\sqrt{2}-1}+\tan ^{-1} \frac{1}{\sqrt{2}}+\frac{21}{10\sqrt{2}}\right ]
Solve :-
\int_0^1 {\frac{{dx}}{{{{({x^2} + 1)}^{3/2}}}} = }

  • 1
  • \frac{1}{{\sqrt 2 }}
  • \frac{1}{2}
  • \sqrt 2
Solve:-
\int\limits_0^1 {\frac{{dx}}{{{{({x^2} + 1)}^{3/2}}}}}
  • 1
  • \sqrt 2
  • 1/2
  • \dfrac{1}{\sqrt 2}
If I= \displaystyle \int_{0}^{1/\sqrt{3}}\displaystyle \frac{dx}{\left ( 1+x^{2} \right )\sqrt{1-x^{2}}} then I is equal to
  • \pi /2
  • \pi /2\sqrt{2}
  • \pi /4\sqrt{2}
  • \pi /4
\displaystyle \int_{0}^{\pi /3}\frac{\cos \theta }{3+4\sin \theta }d\theta =\lambda \log \frac{3+2\sqrt{3}}{3} then \displaystyle \lambda equals
  • \dfrac{1}{2}
  • \dfrac{1}{3}
  • \dfrac{1}{4}
  • \dfrac{1}{8}
The value of \displaystyle \int ^{\tan x}_{1/e}\displaystyle \frac{t\, dt}{1+t^{2}}+\displaystyle \int ^{\cot x}_{1/e}\displaystyle \frac{dt}{t\left ( 1+t^{2} \right )} is
  • \displaystyle \frac{1}{2+tan^{2}x}
  • 1
  • \pi /4
  • \displaystyle \frac{2}{\pi }\displaystyle \int ^{1}_{-1}\displaystyle \frac{dt}{1+t^{2}}
If \displaystyle I_{n}= \int_{0}^{\pi /4}\tan ^{n}x\times \sec ^{2}x dx, then \displaystyle I_{1}, I_{2}, I_{3}, .......are in
  • A.P.
  • G.P.
  • H.P.
  • none
The value of the integral \displaystyle \int_{\alpha }^{\beta }\displaystyle \dfrac{dx}{\sqrt{\left ( x-\alpha  \right )\left ( \beta -x \right )}} for \beta > \alpha , is
  • \sin ^{-1}\: \alpha /\beta
  • \pi /2
  • \sin ^{-1}\beta /2\alpha
  • \pi
\displaystyle \int_{\pi /6}^{\pi /4}\frac{dx}{\sin 2x}is equal to
  • \displaystyle \frac{1}{2}\log \left ( -1 \right )
  • \displaystyle \log \left ( -1 \right )
  • \displaystyle \log 3
  • \displaystyle \frac {1}{2} \log \sqrt{3}
Value of \displaystyle \int_{0}^{1}\displaystyle \frac{dx}{\left ( 1+x^{2} \right )\sqrt{1-x^{2}}} is?
  • \dfrac{\pi}{2\sqrt{2}}
  • \dfrac{\pi}{\sqrt{2}}
  • \sqrt{2}\pi
  • 2\sqrt{2\pi }
If 3+2\displaystyle\int _{ 0 }^{ 1 }{ { x }^{ 2 }{ e }^{ -x^{ 2 } } } dx=\displaystyle\int _{ 0 }^{ 1 }{ { e }^{ -x^{ 2 } } } dx then the value of \beta is
  • \dfrac{1}{e}
  • e
  • \dfrac{1}{2e}
  • 2e
If I_n=\displaystyle\int^1_0\dfrac{dx}{(1+x^2)^n}; n\in N, then which of the following statements hold good?
  • 2n I_{n+1}=2^{-n}+(2n-1)I_n
  • I_2=\dfrac{\pi}{8}+\dfrac{1}{4}
  • I_2=\dfrac{\pi}{8}-\dfrac{1}{4}
  • I_3=\dfrac{\pi}{16}-\dfrac{5}{48}
If \int { \frac { { x }^{ 1/2 } }{ { x }^{ 1/2 }-{ x }^{ 1/3 } }  } dx=Ax\frac { 6 }{ 5 } { x }^{ 5/6 }+\frac { 3 }{ 2 } { x }^{ 2/3 }+B\sqrt { x } +{ Cx }^{ 1/3 }+{ Dx }^{ 1/6 }+E In \left( { x }^{ 1/6 }-1 \right) =k, then A+B+C+D+E=
  • 6
  • 12
  • 18
  • 17
The value of \displaystyle \int_{1/2}^{1}\displaystyle \frac{dx}{x\sqrt{3x^{2}+2x-1}} is?
  • \pi /2
  • \pi /3
  • \pi /6
  • \pi /\sqrt{2}
If I= \displaystyle \int_{1}^{\infty }\displaystyle \frac{x^{2}-2}{x^{3}\sqrt{x^{2}-1}}\: dx, then I equals
  • -1
  • 0
  • \pi /2
  • \pi -\sqrt{3}
If 0< \alpha < \pi /2 then the value of \displaystyle \int_{0}^{\alpha }\displaystyle \frac{dx}{1-\cos x\cos \alpha } is
  • \pi /\alpha
  • \pi /2\sin \alpha
  • \pi /2\cos \alpha
  • \pi /2\alpha
Value of \displaystyle \int_{a}^{\infty }\displaystyle \frac{dx}{x^{4}\sqrt{a^{2}+x^{2}}} is
  • \displaystyle \frac{2+\sqrt{2}}{3a^{4}}
  • \displaystyle \frac{2-\sqrt{2}}{3a^{2}}
  • \displaystyle \frac{2-\sqrt{2}}{3a^{4}}
  • \displaystyle \frac{\sqrt{2}+1}{3a^{2}}
The value of \displaystyle \int_{-4}^{-5}e^{\left ( x+5 \right )^{2}}dx+3\displaystyle \int_{1/3}^{2/3}e^{9\left ( x-2/3 \right )^{2}}dx is
  • 2/5
  • 1/5
  • 1/2
  • none of these
Value of \displaystyle \int_{0}^{16}\displaystyle \frac{x^{1/4}}{1+x^{1/2}}\: dx is
  • \displaystyle \frac{8}{3}
  • \displaystyle \frac{4}{3}\tan ^{-1}\: 2
  • 4\displaystyle \left ( \displaystyle \frac{2}{3}+\tan ^{-1}\: 2 \right )
  • 4\displaystyle \left ( \displaystyle \frac{2}{3}-\tan ^{-1}\: 2 \right )
The solution of x of the equation \displaystyle \int_{\sqrt{2}}^{x}{\dfrac{dt}{t\sqrt{t^{2}-1}}}=\dfrac{\pi}{2} is 
  • 2\sqrt{2}
  • 2
  • \pi
  • -\sqrt{2}
\int_{0}^{1}\frac{1}{ax+b}dx is
  • zero
  • log_{e}\frac{a+b}{b}
  • log_{e}(ax+b)
  • \frac{1}{a}log_{e}\left ( \frac{a+b}{b} \right )
\int _{ 0 }^{ 8 }{ \left[ \sqrt { t }  \right]  } dt at equals to (where [.] greatest integer function.)
  • 28
  • 11
  • 2
  • 8
If \displaystyle \frac{\pi }{4}< \alpha < \displaystyle \frac{\pi }{2}, value of \displaystyle \int_{-\pi /2}^{\pi /2}\displaystyle \frac{\sin 2x}{\sqrt{1+\sin 2\alpha \sin x}} is
  • -\displaystyle \frac{4}{3}\tan \alpha \sec \alpha
  • -\displaystyle \frac{4}{3}\cot \alpha cosec\alpha
  • -\displaystyle \frac{4}{3}\tan \alpha cosec\alpha
  • -\displaystyle \frac{4}{3}\cot \alpha \sec \alpha
If I_{1}= \displaystyle \int_{x}^{1}\displaystyle \frac{dt}{1+t^{2}} and I_{2}= \displaystyle \int_{1}^{1/x}\displaystyle \frac{dt}{1+t^{2}} for x> 0, then
  • I_{1}= I_{2}
  • I_{1}> I_{2}
  • I_{2}> I_{1}
  • I_{2}= \left ( \pi /2 \right )-\tan ^{-1}x
If I_{1}= \displaystyle \int_{0}^{\infty }\displaystyle \frac{dx}{1+x^{4}} and I_{2}= \displaystyle \int_{0}^{\infty }\displaystyle \frac{x^{2}}{1+x^{4}}\: dx, then
  • I_{1}= I_{2}
  • I_{1}=2 I_{2}
  • 2I_{1}= I_{2}
  • none of these
If \displaystyle \int_{0}^{\infty }\displaystyle \frac{\log \left ( 1+x^{2} \right )}{1+x^{2}}\: dx= \lambda \displaystyle \int_{0}^{1}\displaystyle \frac{\log \left ( 1+x \right )}{1+x^{2}}\: dx then \lambda  equals
  • 4
  • \pi
  • 8
  • 2\pi
0:0:1


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