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CBSE Questions for Class 12 Commerce Maths Integrals Quiz 8 - MCQExams.com

552(25x2)3x4dx is equal to
  • π6
  • 2π3
  • 5π6
  • π3
The minimum value of the function f(x) = \int^x_0 \frac{d \theta}{cos \theta} + \int^{\pi/2}_x \frac{d \theta}{sin \theta} where x \in [0, \frac{\pi}{2}], is  
  • 2ln(\sqrt{2} + 1)
  • ln(2\sqrt{2} + 2)
  • ln(\sqrt{3} + 2)
  • ln(\sqrt{2} + 3)
If  \displaystyle \int_{0}^{1} \frac{\tan^{-1} x}{x}dx  is equal to
  • \displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sin x}{x}dx
  • \displaystyle \int_{0}^{\frac{\pi}{2}}\frac{x}{\sin x}dx
  • \displaystyle \frac{1}{2} \int_{0}^{\frac{\pi}{2}}\frac{\sin x}{x}dx
  • \displaystyle \frac{1}{2} \int_{0}^{\frac{\pi}{2}}\frac{x}{\sin x}dx
The value of the integral \displaystyle \int_{0 }^{\log5}\frac{e^x\sqrt{e^x-1}}{e^x+3}\:dx is
  • 3+2\pi
  • 4-\pi
  • 2+\pi
  • none of these
The value of the integral  \displaystyle \int_{0}^{\frac{1}{\sqrt{3}}}\frac{dx}{(1+x^2)\sqrt{1-x^2}}
  • \displaystyle \frac{\pi }{2\sqrt{2}}
  • \displaystyle \frac{\pi }{4\sqrt{2}}
  • \displaystyle \frac{\pi }{8\sqrt{2}}
  • none of these
Let   \displaystyle I_1=\int_{0}^{1}\frac{e^x dx}{1+x} and \displaystyle I_2=\int_{0}^{1}\dfrac{x^2 dx}{e^{x^3}(2-x^3)}.
Then \dfrac{I_1}{I_2} is equal to?
  • \displaystyle \dfrac{3}{e}
  • \displaystyle \frac{e}{3}
  • \displaystyle {3}{e}
  • \displaystyle \dfrac{1}{3e}
The value of \displaystyle \int_{0}^{\infty} \frac {dx}{1 + x^4} is
  • same as that of \displaystyle \int_{0}^{\infty} \frac {x^2 + 1dx}{1 + x^4}
  • \displaystyle \frac {\pi}{2\sqrt{2}}
  • same as that of \displaystyle \int_{0}^{\infty} \frac {x^2 \: dx}{1 + x^4}
  • \displaystyle \frac {\pi}{\sqrt{2}}
Evaluate \displaystyle\int_0^{\displaystyle\frac{\pi}{4}}{\frac{\sin{x}+\cos{x}}{9+16\{1-{(\sin{x}-\cos{x})}^2\}}dx}
  • \displaystyle\frac{1}{20}(\log{\sqrt{3}})
  • \displaystyle\frac{1}{10}(\log{3})
  • \displaystyle\frac{1}{20}(\log{3})
  • \displaystyle\frac{1}{20}(\log{3}-\log{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
Let \displaystyle I_1 = \int_1^2 \frac{1}{\sqrt{1 + x^2}} dx and I_2 \displaystyle = \int_1^2 \frac{1}{x} dx. Then
  • I_1 > I_2
  • I_2 > I_1
  • I_1 = I_2
  • I_1 > 2 I_2
Let \displaystyle\frac{d}{dx}(F(x))=\frac{e^{\displaystyle\sin{x}}}{x}, x>0. If \displaystyle\int_1^4{\frac{2e^{\displaystyle\sin{x^2}}}{x}dx}=F(k)-F(1), then the possible value of k is
  • 10
  • 14
  • 16
  • 18
The evaluation of \displaystyle \int \frac{pX^{p+2q-1}-qX^{q-1}}{X^{2p+2q}+2X^{p+q}+1}dx is 
  • -\displaystyle \frac{X^p}{X^{p+q}+1}+C
  • \displaystyle \frac{X^q}{X^{p+q}+1}+C
  • -\displaystyle \frac{X^q}{X^{p+q}+1}+C
  • \displaystyle \frac{X^p}{X^{p+q}+1}+C
Let \displaystyle \frac{df\left ( x \right )}{dx}=\frac{e^{\sin x}}{x}, x> 0. If \displaystyle \int_{1}^{4}\displaystyle \frac{3e^{\sin x^{3}}}{x}dx=f\left ( k \right )-f\left ( 1 \right ) then one of the possible values of k is
  • 16
  • 63
  • 64
  • 15
\displaystyle \int_{1}^{e^{37}}\frac{\pi \sin \left ( \pi \log _{e}x \right )}{x}dx is equal to
  • 2
  • \displaystyle -2
  • \displaystyle 2/\pi
  • \displaystyle 2\pi
\displaystyle \int_{0}^{2}\sqrt{\frac{2+x}{2-x}}dx is equal to
  • \displaystyle \pi +1
  • \displaystyle 1+\pi /2
  • \displaystyle \pi +3/2
  • none of these
\displaystyle\int_0^\infty{f\left(x+\frac{1}{x}\right).\frac{\ln{x}}{x}dx} is equal to
  • 0
  • 1
  • \displaystyle\frac{1}{2}
  • cannot be evaluated
 \displaystyle \int_{1}^{\infty }\frac{\log \left ( t-1 \right )}{t^2\log t+\log \left ( \frac{t}{t-1} \right )}\:dt equals
  • \displaystyle \frac{1}{2}
  • \displaystyle \frac{1}{3}
  • \displaystyle \frac{2}{3}
  • 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 \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}}
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 }  }
\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
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
\displaystyle\int_{0}^{a}x^{4}\left ( a^{2}-x^{2} \right )^{1/2} dx equals
  • \displaystyle\frac{\pi a^{5}}{32}
  • \displaystyle\frac{\pi a^{6}}{32}
  • \displaystyle \frac{\pi a^{2}}{32}
  • None of these
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
\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 )
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 ]
If \displaystyle \int_{\log 2}^{x}\displaystyle \frac{dx}{\sqrt{e^{x}-1}}= \displaystyle \frac{\pi }{6}, the value of x is
  • 4
  • \log 8
  • \log 4
  • none of these
\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}}
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 }
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
If \displaystyle \int _{ 0 }^{ 1 }{ \frac { \sin { t }  }{ 1+t } dt } =\alpha , then the value of the integral \displaystyle \int _{ 4\pi -2 }^{ 4\pi  }{ \frac { \sin { t/2 }  }{ 4\pi +2-t } dt } in terms of \alpha is given by
  • 2\alpha
  • -2\alpha
  • \alpha
  • -\alpha
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 )
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
 If \displaystyle I=\int _{8}^{15} \frac{dx}{(x-3)\sqrt{x+1} } then I equals
  • \displaystyle \frac{1}{2}\log \frac {5}{3}
  • \displaystyle 2 \log \frac{1}{3}
  • \displaystyle \frac{1}{2}-\log \frac {1}{5}
  • \displaystyle 2 \log \frac{5}{3}
37 If n > 1, and \displaystyle I=\int _{0}^{\infty} \frac{dx}{(x+\sqrt{1+x^{2}})^{n}} then I  equals
  • \displaystyle \frac{n}{n^{2}-1}
  • \displaystyle \frac{2n}{n^{2}-1}
  • \displaystyle \frac{n}{2(n^{2}-1)}
  • \displaystyle \sqrt{ n^{2}-1}
A function f is defined by \displaystyle f(x)=\frac{1}{2^{r-1}},\frac{1}{2r}<x\leq \frac{1}{2^{r-1}},r=1,2,3,..... then the value of \displaystyle \int _{0}^{1}f(x)dx is equal
  • \cfrac 13
  • \cfrac 14
  • \cfrac 23
  • \cfrac 12
0:0:1


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