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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) = x0dθcosθ+π/2xdθsinθ where x[0,π2], is  
  • 2ln(2+1)
  • ln(22+2)
  • ln(3+2)
  • ln(2+3)
If 10tan1xxdx  is equal to
  • π20sinxxdx
  • π20xsinxdx
  • 12π20sinxxdx
  • 12π20xsinxdx
The value of the integral log50exex1ex+3dx is
  • 3+2π
  • 4π
  • 2+π
  • none of these
The value of the integral  130dx(1+x2)1x2
  • π22
  • π42
  • π82
  • none of these
Let  I1=10exdx1+x and I2=10x2dxex3(2x3).
Then I1I2 is equal to?
  • 3e
  • e3
  • 3e
  • 13e
The value of 0dx1+x4 is
  • same as that of 0x2+1dx1+x4
  • π22
  • same as that of 0x2dx1+x4
  • π2
Evaluate π40sinx+cosx9+16{1(sinxcosx)2}dx
  • 120(log3)
  • 110(log3)
  • 120(log3)
  • 120(log3log2)
If 2f(x)+f(x)=1xsin(x1x), then the value of e1ef(x)dx, is
  • 1
  • 0
  • e
  • 1
Let I1=2111+x2dx and I2=211xdx. Then
  • I1>I2
  • I2>I1
  • I1=I2
  • I1>2I2
Let ddx(F(x))=esinxx, x>0. If 412esinx2xdx=F(k)F(1), then the possible value of k is
  • 10
  • 14
  • 16
  • 18
The evaluation of pXp+2q1qXq1X2p+2q+2Xp+q+1dx is 
  • XpXp+q+1+C
  • XqXp+q+1+C
  • XqXp+q+1+C
  • XpXp+q+1+C
Let df(x)dx=esinxx,x>0. If 413esinx3xdx=f(k)f(1) then one of the possible values of k is
  • 16
  • 63
  • 64
  • 15
e371πsin(πlogex)xdx is equal to
  • 2
  • 2
  • 2/π
  • 2π
202+x2xdx is equal to
  • π+1
  • 1+π/2
  • π+3/2
  • none of these
0f(x+1x).lnxxdx is equal to
  • 0
  • 1
  • 12
  • cannot be evaluated
1log(t1)t2logt+log(tt1)dt equals
  • 12
  • 13
  • 23
  • None of these
If It=π20sin2txsin2xdx then ,I1,I2,I3 are in
  • A.P.
  • H.P.
  • G.P.
  • None of these
The value of π/20sinθlog(sinθ)dθ equals
  • loge(1e)
  • log2e
  • loge21
  • loge(e2)
If I=π1/π1xsin(x1x)dx then I is equal to
  • 0
  • π
  • π1π
  • π+1π
If x satisfies the equation (10dtt2+2tcosα+1)x2(33t2sin2tt2+1dt)x2=0 
for (0<α<π)
then the value of x is?
  • ±α2sinα
  • ±2sinαα
  • ±αsinα
  • ±2sinαα
Let F(x)=f(x)+f(1x) where f(x)=x1logt1+tdt 
Then F(e) is equal to?
  • 1
  • 2
  • 1/2
  • 0
The value of π/20dθ5+3cosθ is?
  • tan112
  • tan113
  • 12tan112
  • 13tan113
011+xndx,n>1 is equal to?
  • 2011+xdx
  • 11+xndx
  • 1dx(xn1)1/n
  • 101(1xn)1/n
If I1=π/20xsinxdx and I2=π/20tan1xxdx, then I1I2= 

  • 12
  • 1
  • 2
  • π2
If11g(x)1+t2dt=f(x),where,g(x)=sinx , then f(π3) equals

  • π4
  • does not exist
  • π34
  • None of these
If In=π40tannxdx 

then 1I2+I4,1I3+I5,1I4+I6 are in?
  • A.P
  • H.P
  • G.P
  • None of these
a0x4(a2x2)1/2dx 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
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