CBSE Questions for Class 12 Commerce Applied Mathematics Indefinite Integrals Quiz 5 - MCQExams.com

$$\displaystyle \int x^{3}\tan ^{-1}x\:dx.$$
  • $$\displaystyle \frac{1}{2}\left [ \left ( x^{4}-1 \right )\tan ^{-1}x-\frac{x^{2}}{3}+x \right ].$$
  • $$\displaystyle \frac{1}{2}\left [ \left ( x^{4}+1 \right )\tan ^{-1}x-\frac{x^{2}}{3}+x \right ].$$
  • $$\displaystyle \frac{1}{4}\left [ \left ( x^{4}-1 \right )\tan ^{-1}x-\frac{x^{2}}{3}+x \right ].$$
  • $$\displaystyle \frac{1}{4}\left [ \left ( x^{4}-1 \right )\tan ^{-1}x-\frac{x^{1}}{3}+x \right ].$$
$$\displaystyle\int { x\left( f\left( { x }^{ 2 } \right) g''\left( { x }^{ 2 } \right) -f''\left( { x }^{ 2 } \right) g\left( { x }^{ 2 } \right)  \right) dx } =$$
  • $$\displaystyle f\left( { x }^{ 2 } \right) g'\left( { x }^{ 2 } \right) -g\left( { x }^{ 2 } \right) f'\left( { x }^{ 2 } \right) +c$$
  • $$\displaystyle \frac { 1 }{ 2 } \left( f\left( { x }^{ 2 } \right) g\left( { x }^{ 2 } \right) f'\left( { x }^{ 2 } \right)  \right) +c$$
  • $$\displaystyle \frac { 1 }{ 2 } \left( f\left( { x }^{ 2 } \right) g'\left( { x }^{ 2 } \right) -g\left( { x }^{ 2 } \right) f'\left( { x }^{ 2 } \right)  \right) +c$$
  • none of these
If $$\displaystyle f\left ( x \right )=\int\frac{1}{x-\sqrt{x^{2}+1}}$$ and $$\displaystyle f\left ( 0 \right )=\frac{1+\sqrt{2}}{2}$$, then $$f(1)$$ is equal to
  • $$\displaystyle \log \left (\sqrt{ \sqrt{2}-1} \right )$$
  • $$\displaystyle \frac {-1}{\sqrt {2}}$$
  • $$\displaystyle 1+ \sqrt{2}$$
  • $$\displaystyle \dfrac12\log \left ( 1+\sqrt{2} \right )$$
If $$\displaystyle I = \int \frac {\log (\cos x)}{\cos^2 x}dx$$, then $$I$$ equals
  • $$\displaystyle \tan x \log \cos x + \tan x - x + C$$
  • $$\displaystyle \tan x \log \cos x - \tan x - x^2 + C$$
  • $$\displaystyle \tan x \log \cos x - \cot x + x + C$$
  • $$\displaystyle \tan x \log \cos x + \cot x - x + C$$
$$\displaystyle \int e^{\sin x}\left ( \frac{x\cos ^{3}x-\sin x}{\cos ^{2}x} \right )dx$$
  • $$\displaystyle xe^{\sin x}-e^{\sin x}.\sec x$$
  • $$\displaystyle e^{\sin x}-e^{\sin x}.\sec x$$
  • $$\displaystyle xe^{\cos x}-e^{\sin x}.\sec x$$
  • $$\displaystyle xe^{\sin x}+e^{\sin x}.\sec x$$
If $$\displaystyle I = \int \cos \theta \log (\tan \theta/2) d\theta$$, then I equals
  • $$\displaystyle \sin \theta \log (\tan \theta/2) + \theta + C$$
  • $$\displaystyle \cos \theta \log (\tan \theta/2) + \theta + C$$
  • $$\displaystyle \sin \theta \log (\tan \theta/2) - \theta + C$$
  • none of these
If $$\displaystyle \int \log \left ( x^{2}+x \right )dx=x\log x+\left ( x+1 \right )\log \left ( x+1 \right )+K,$$ then K is equal to
  • $$\displaystyle 2x+\log\left ( x+1 \right )+C$$
  • $$\displaystyle 2x-\log\left ( x+1 \right )+C$$
  • constant
  • none of these
$$\displaystyle \int e^{\cos x}\frac{\left (  x\sin^{3}x+\cos x \right )}{\sin ^{2}x}dx$$
  • $$\displaystyle x.e^{\cos x}-e^{\cos x}co\sec x.$$
  • $$\displaystyle -.e^{\cos x}+e^{\cos x}co\sec x.$$
  • $$\displaystyle -x.e^{\cos x}-e^{\cos x}co\sec x.$$
  • $$\displaystyle -x.e^{\cos x}+e^{\cos x}co\sec x.$$
$$\displaystyle \int { { 2 }^{ x }\left( f'(x)+f(x)\log { 2 }  \right) dx } $$ is
  • $${ 2 }^{ x }f'(x)+C$$
  • $${ 2 }^{ x }f(x)+C$$
  • $${ 2 }^{ x }\left( \log { 2 } \right) f(x)+C$$
  • $$\left( \log { 2 } \right) f(x)+C$$

If the graph of the antiderivative F(x) of f(x) = log (logx) + $$(logx)^{-2}$$ passes through (e. 1998\, -\, e) then the term independent of x in F(x) is .......... .
  • 1998
  • 1999
  • 1997
  • 1996
If $$\displaystyle \int { \frac { { x }^{ 2 } }{ { \left( x\sin { x } +\cos { x }  \right)  }^{ 2 } } dx } =\frac { f\left( x \right) }{ x\sin { x } +\cos { x }  } +\tan { x } +c$$, then
  • $$\displaystyle f\left( x \right)=\frac { x }{ \cos { x }  } $$
  • $$\displaystyle f\left( x \right) =\frac { \cos { x }  }{ x } $$
  • $$\displaystyle f\left( x \right)=-\frac { x }{ \cos { x }  } $$
  • none of these
$$\int\displaystyle\frac{x^2+1}{x^4+1}dx$$ is equal to.
  • $$\displaystyle\frac{1}{\sqrt 2}\tan^{-1}\left(\displaystyle\frac{x^2}{\sqrt 2 x}\right)+c$$
  • $$\displaystyle\frac{1}{\sqrt 2}\tan^{-1}\left(\displaystyle\frac{x^2-1}{\sqrt 2 x}\right)+c$$
  • $$\displaystyle\frac{1}{\sqrt 2}\tan^{-1}\left(\displaystyle\frac{x^2+1}{x}\right)+c$$
  • $$\displaystyle\tan^{-1}\left(\displaystyle\frac{x^2-1}{\sqrt 2 x}\right)+c$$
Evaluate $$\displaystyle \int xe^{x} dx$$.
  • $$-xe^{x} - e^{x} + c$$
  • $$-xe^{x} + e^{x} + c$$
  • $$xe^{x} + e^{x} + c$$
  • $$xe^{x} - e^{x} + c$$
Identify the correct expression :
  • $$x \int lnx\, dx = x^{2} ln |x| - x^{2} + c$$
  • $$x \int ln |x| dx = xe^{x} + c$$
  • $$x \int e^{x} dx = xe^{x} + c x$$
  • $$\displaystyle \int \frac{dx}{\sqrt{a^{2} + x^{2}}} = \frac{1}{a} tan^{-1} \left ( \frac{x}{a} \right ) + c$$
$$\displaystyle \int{\tan^2x}dx$$
  • $$\tan x-x+c$$
  • $$\tan x+c$$
  • $$\tan x-x$$
  • None of the above
Primitive of $$\displaystyle \frac{3x^4\, -\, 1}{(x^4\, +\, x\, +\, 1)^2}$$ w.r.t. x is
  • $$\displaystyle \frac{x}{x^4\, +\, x\, +\, 1}\, +\, c$$
  • $$\displaystyle - \frac{x}{x^4\, +\, x\, +\, 1}\, +\, c$$
  • $$\displaystyle \frac{x\, +\, 1}{x^4\, +\, x\, +\, 1}\, +\, c$$
  • $$\displaystyle -\frac{x\, +\, 1}{x^4\, +\, x\, +\, 1}\, +\, c$$
Evaluate $$\displaystyle \int (\tan(e^{x}) + xe^{x}. \sec^{2}(e^{x}))dx$$
  • $$x\, -\tan(e^{x}) + c$$
  • $$x\, \tan(e^{x}) + c$$
  • $$x\, \tan(e^{-x}) + c$$
  • $$x\, -\tan(e^{-x}) + c$$
If $$\dfrac {dy}{dt} = ky$$ and $$k\neq 0$$, which of the following could be the equation of $$y$$?
  • $$y = kx - 7$$
  • $$y = 95e^{kt}$$
  • $$y = 5 + ln\ k$$
  • $$y = (x - k)^{2}$$
  • $$y = \sqrt [k]{x}$$
$$\displaystyle \int (1 + x - x^{-1})e^{x + x^{-1}}dx$$ is equal to
  • $$(x + 1)e^{x + x^{-1}} + C$$
  • $$(x - 1)e^{x + x^{-1}} + C$$
  • $$xe^{x + x^{-1}} + C$$
  • $$xe^{x + x^{-1}} x + C$$
If $$\displaystyle \int { \cfrac { f(x) }{ \log { (\sin { x } ) }  }  } dx=\log { \left[ \log { \sin { x }  }  \right]  } +c$$, then $$f(x)=$$........
  • $$\cot { x } $$
  • $$\tan { x } $$
  • $$\sec { x } $$
  • $$co\sec { x } $$
$$\int \dfrac {1}{x^{2} (x^{4} + 1)^{3/4}}dx$$ is equal to
  • $$\left (1 + \dfrac {1}{x^{4}}\right )^{1/4} + C$$
  • $$(x^{4} + 1)^{1/4} + C$$
  • $$\left (1 - \dfrac {1}{x^{4}}\right )^{1/4} + C$$
  • $$-\left (1 + \dfrac {1}{x^{4}}\right )^{1/4} + C$$
$$\displaystyle \int \frac{\sin 2x}{\sin^2x+2 \cos^2x}dx=$$
  • $$\log (1+cos^2x)+C$$
  • $$\log (1+tan^2x)+C$$
  • $$-\log (1+sin^2x)+C$$
  • $$-\log (1+cos^2x)+C$$
What is $$\displaystyle \int \dfrac{x^4 - 1}{x^2 \sqrt{x^4 + x^2 + 1}} dx$$ equal to?
  • $$\sqrt{\dfrac{x^4 + x^2 + 1}{x}} + c$$
  • $$\sqrt{x^4 + 2 - \dfrac{1}{x^2}} + c$$
  • $$\sqrt{x^2 + \dfrac{1}{x^2} + 1} + c$$
  • $$\sqrt{\dfrac{x^4 - x^2 + 1}{x}} + c$$
What is $$\displaystyle\int { \left( x\cos { x } +\sin { x }  \right) dx } $$ equal to?
Where $$c$$ is an arbitrary constant
  • $$x\sin { x } +c$$
  • $$x\cos { x } +c$$
  • $$-x\sin { x } +c$$
  • $$-x\cos { x } +c$$
$$\int { \left( \cfrac { 4{ e }^{ x }-25 }{ 2{ e }^{ x }-5 }  \right)  } dx=Ax+B\log { \left| 2{ e }^{ x }-5 \right|  } +c$$, then
  • $$A=5,B=3$$
  • $$A=5,B=-3$$
  • $$A=-5,B=3$$
  • $$A=-5,B=-3$$
The value of $$\displaystyle  \int \dfrac {e^x}{x} ( x \log x+1) \  dx $$  is equal to 
  • $$ \dfrac {e^x} {x} + C $$
  • $$ x e^x \log |x| + C $$
  • $$ e^x \log |x| + C $$
  • $$ x ( e^x + \log |x| ) + C $$
  • $$ xe^x + \log |x| + C $$
$$\displaystyle\int \displaystyle\frac{\cos \alpha}{\sin x\cos (x-\alpha)}dx=$$___________ $$+$$c where $$0 < x < \alpha < \pi_{/2}$$ and $$\alpha$$-constant.
  • $$-ln|tan x+\cot \alpha|$$
  • $$ln|\cot x+\tan\alpha|$$
  • $$ln|tan x+\cot \alpha|$$
  • $$-ln|\cot x+\tan\alpha|$$
If $$\int \dfrac {1}{x \sqrt {1 - x^{3}}}dx = a\log \left |\dfrac {\sqrt {1 - x^{3}} - 1}{\sqrt {1 - x^{3}} + 1}\right | + b$$ then $$a =$$
  • $$\dfrac {2}{3}$$
  • $$\dfrac {1}{3}$$
  • $$-\dfrac {2}{3}$$
  • None of these
$$\int e^{-x} \text{cosec} x(1 + \cot x) dx$$ is equal to
  • $$-e^{-x} \text{cosec} x + C$$
  • $$e^{-x} \text{cosec} x + C$$
  • $$-e^{-x} (\text{cosec} x + \cot x) + C$$
  • $$-e^{-x} (\text{cosec} x + \tan x) + C$$
  • $$-e^{-x}\sec x + C$$
The solution of the equation $$\dfrac { dy }{ dx } =\dfrac { x\left( 2\log { x } +1 \right)  }{ \sin { y } +y\cos { y }  } $$ is
  • $$y\sin { y } ={ x }^{ 2 }\log { x } +\dfrac { { x }^{ 2 } }{ y } +c$$
  • $$y\cos { y } ={ x }^{ 2 }\left( \log { x } +1 \right) +c$$
  • $$y\cos { y } ={ x }^{ 2 }\log { x } +\dfrac { { x }^{ 2 } }{ 2 } +c$$
  • $$y\sin { y } ={ x }^{ 2 }\log { x } +c$$
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Practice Class 12 Commerce Applied Mathematics Quiz Questions and Answers