CBSE Questions for Class 12 Commerce Applied Mathematics Indefinite Integrals Quiz 8 - MCQExams.com

The value of $$\int \dfrac {1}{\sqrt {\sin^{3}x \cos^{5}x}} dx$$ is
  • $$\dfrac {-2}{\sqrt {\tan x}} + \dfrac {2}{3} (\tan x)^{3/2} + C$$
  • $$\dfrac {2}{\sqrt {\tan x}} - \dfrac {2}{3} (\tan x)^{3/2} + C$$
  • $$\dfrac {-2}{\sqrt {\tan x}} + \dfrac {2}{3} (\tan x)^{1/2} + C$$
  • None of these
Evaluate :
$$\int x \sin 3x \ dx$$
  • $$-\dfrac{1}{3}x \cos 3x+\dfrac{1}{9} \sin 3x +C$$
  • $$-\dfrac{1}{3} x \cos 3x +\dfrac{1}{3} \sin 3x +C$$
  • $$\dfrac{1}{3} x \sin 3x -\dfrac{1}{3} \cos 3x+C$$
  • None of these
If $$y=(x+\sqrt{x^{2}-a^{2}})^{n}$$ then $$(x^{2}-a^{2})(\dfrac{dy}{dx})^{2}$$=
  • $$n^{2}y$$
  • $$-n^{2}y$$
  • $$ny^{2}$$
  • $$n^{2}y^{2}$$
$$\int e^{x/2}\sin \left( \frac { \pi  }{ 4 } +\frac { x }{ 2 }  \right) dx=$$
  • $$\sqrt{2}\ e^{x/2}\sin \frac { x }{ 2 }+c$$
  • $$\sqrt{2}\ e^{x/2}\cos\frac { x }{ 2 }+c$$
  • $$-\sqrt{2}\ e^{x/2}\sin \frac { x }{ 2 }+c$$
  • $$-\sqrt{2}\ e^{x/2}\cos\frac { x }{ 2 }+c$$
$$\displaystyle \int \dfrac{x \,\ell nx}{(x^2 - 1)^{3/2}}dx$$ equals
  • $$arc \,\sec x - \dfrac{\ell n\,x}{\sqrt{x^2 - 1}} + C$$
  • $$\sec^{-1} x - \dfrac{\ell n\,x}{\sqrt{x^2 - 1}} + C$$
  • $$\cos^{-1} x - \dfrac{\ell n\,x}{\sqrt{x^2 - 1}} + C$$
  • $$\sec x - \dfrac{\ell n\,x}{\sqrt{x^2 - 1}} + C$$
$$\int \dfrac {\sin^{-1}x-\cos^{-1}x}{\sin^{-1}x+\cos^{-1}x}dx=$$
  • $$\dfrac {4}{\pi}[x \sin^{-1}+\sqrt {1-x^{2}}]-x+c$$
  • $$\log [\sin^{-1}+\sqrt {1-x^{2}}]-x+c$$
  • $$\dfrac {4}{\pi}[x \sin^{-1}+\sqrt {1-x^{2}}]+c$$
  • $$\dfrac {2}{\pi}[x \sin^{-1}x-x \cos^{-1}x+2\sqrt {1-x^{2}}]+c$$
$$\int { { x }^{ 2 } } \sqrt { { x }^{ 6 }-1 } dx=$$
  • $$\frac { 1 }{ 6 } \left\{ { x }^{ 3 }\sqrt { { x }^{ 6 }-1 } -\log { \left( { x }^{ 3 }+\sqrt { { x }^{ 6 }-1 } \right) } \right\} +c$$
  • $$\frac { 1 }{ 6 } \left\{ { x }^{ 3 }\sqrt { { x }^{ 6 }-1 } +\log { \left( { x }^{ 3 }+\sqrt { { x }^{ 6 }-1 } \right) } \right\} +c$$
  • $$\frac { 1 }{ 6 } \left\{ { x }^{ 3 }\sqrt { { x }^{ 6 }-1 } -\sin ^{ -1 }{ \left( { x }^{ 3 } \right) } \right\} +c$$
  • $$\frac { 1 }{ 6 } \left\{ { x }^{ 3 }\sqrt { { x }^{ 6 }-1 } +\sin ^{ -1 }{ \left( { x }^{ 3 } \right) } \right\} +c$$
Let $$A = \int _ { 0 } ^ { 1 } \frac { e ^ { t } } { t + 1 }$$ dt,  then the value of $$\int _ { 0 } ^ { 1 } \frac { t e ^ { t ^ { 2 } } } { t ^ { 2 } + 1 } d t$$ is:-
  • $$A ^ { 2 }$$
  • $$\frac { 1 } { 2 } A$$
  • 2$$A$$
  • $$\frac { 1 } { 2 } A ^ { 2 }$$
The value of the integral $$\int _{ -\pi /2 }^{ \pi /2 }{ \left[ { x }^{ 2 }+log\frac { \pi -x }{ \pi +x }  \right]  } $$ cos x dx is 
  • 0
  • $$\frac { { \pi }^{ 2 } }{ 2 } -4$$
  • $$\frac { { \pi }^{ 2 } }{ 2 } +4$$
  • $$\frac { { \pi }^{ 2 } }{ 2 } $$
$$\int _ { 0 } ^ { 1 } \frac { d x } { \sqrt { x + 1 } + \sqrt { x } } d x =$$
  • $$\frac { 4 } { 3 } ( \sqrt { 2 } + 1 )$$
  • $$\frac { 4 } { 3 } ( \sqrt { 2 } - 1 )$$
  • $$\frac { 3 } { 4 } ( \sqrt { 2 } - 1 )$$
  • $$\frac { 3 } { 4 } ( \sqrt { 2 } - 2 )$$
$$\int \dfrac {x+\sin }{1+\cos x}dx=$$
  • $$x\ \tan \dfrac {x}{2}+c$$
  • $$x\ \cot \dfrac {x}{2}$$
  • $$x\ \sin\dfrac {x}{2}+c$$
  • $$x\ \cos \dfrac {x}{2}$$
Solve $$\displaystyle \int { x\sin ^{ 2 }{ x }  } dx$$ 
  • $$\dfrac{(x-1)}{2}(x-\dfrac{cos2x}{2})+C$$ 
  • $$\dfrac{(x-1)}{2}(x-\dfrac{sin2x}{2})+C$$ 
  • $$\dfrac{(x+1)}{2}(x-\dfrac{sin2x}{2})+C$$ 
  • $$None\ of\ these$$
If $$\displaystyle\int {\dfrac{1}{{1 + \sin x}}dx = \tan \left( {\frac{x}{2} + a} \right) + b} $$, then 
  • $$a = - \dfrac{\pi }{4},b \in R$$
  • $$a = \dfrac{\pi }{4},b \in R$$
  • $$a = \dfrac{{5\pi }}{4},b \in R$$
  • None of these
Solve $$\int {{x^3}\log xdx  } $$
  • $$\dfrac{{{x^4}\log x}}{4} + C$$
  • $$ I=\dfrac{{{x}^{4}}}{4}\log x-\dfrac{{{x}^{4}}}{16}+C $$
  • $$\dfrac{1}{{8}}\left[ {{x^4}\log x - 4{x^2}} \right] + C$$
  • $$\dfrac{1}{{16}}\left[ {4{x^4}\log x + {x^4}} \right] + C$$
Evaluate :
$$\displaystyle\int {\dfrac{9cosx-sinx}{4sinx+5cosx}dx}$$
  • $$x+\ln(4\sin x+\cos x)+c$$
  • $$x+\ln(\sin x-5\cos x)+c$$
  • $$x+\ln(4\sin x+5\cos x)+c$$
  • None of these
If $$\int { \cfrac { \sin { x }  }{ \sin { \left( x-\alpha  \right)  }  }  } dx=Ax+B\log { \sin { \left( x-\alpha  \right)  }  } +c$$, then the value of $$(A,B)$$ is-
  • $$\left( \sin { \alpha } ,\cos { \alpha } \right) $$
  • $$\left( \cos { \alpha } ,\sin { \alpha } \right) $$
  • $$\left( -\sin { \alpha } ,\cos { \alpha } \right) $$
  • $$\left(- \cos { \alpha } ,\sin { \alpha } \right) $$
Solve $$\displaystyle \int_{1}^{2}\dfrac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}}\ dx$$
  • $$1$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{3}{2}$$
  • 0
If $$f(3-x)=f(x)$$, then $$\int_{1}^{2}x\ f(x)\ dx$$ is equal to
  • $$\dfrac{3}{2}\int_{1}^{2}f(2-x)\ dx$$
  • $$\dfrac{3}{2}\int_{1}^{2}f(x)\ dx$$
  • $$\dfrac{1}{2}\int_{1}^{2}f(x)\ dx$$
  • $$\int_{1}^{2}f(x)\ dx$$
If $$f(x)=\int { \left( \dfrac { x^2+\sin^2x }{ 1+x^2 }  \right)  } \sec^2 x dx $$ and f(0)=0,then  f (1) equals:
  • $$1-\dfrac { \pi }{ 4 } $$
  • $$-\dfrac { \pi }{ 4 } $$
  • $$\tan 1-\dfrac { \pi }{ 4 } $$
  • $$\tan 1+1$$
The value of $$\int$$ $$(x e^{ln sinx} - cos x)$$ dx is equal to :
  • -x cos x + C
  • sin x - x cos x + C
  • $$- e^{lnx} cos x + C$$
  • sin x + x cos x + C
The integrating factor of the differential equation $$\dfrac{dy}{dx}\left(x\log _e\:x\right)+y=2\log _e\:x$$ is given by
  • $$x$$
  • $$e^x$$
  • $$\log _e\:x$$
  • $$\log _e\left(\log _e\:x\right)$$
$$\displaystyle \int \dfrac{\{f(x) \cdot \phi' (x) - f'(x) \cdot \phi(x) \}}{f(x) \cdot \phi(x)} \{log \,\phi(x) - log \,f(x) \}dx$$ is equal to
  • $$log \dfrac{\phi(x)}{f(x)} + k$$
  • $$\dfrac{1}{2} \left \{log \dfrac{\phi(x)}{f(x)} \right \}^2 + k$$
  • $$\dfrac{\phi(x)}{f(x)} log \dfrac{\phi(x)}{f(x)} + k$$
  • None of these
$$\int \{1 + 2 \tan x (\tan x + \sec x)\}^{\dfrac{1}{2}} dx =$$
  • $$log (\sec x + \tan x) + c$$
  • $$log (\sec x + \tan x)^{\dfrac{1}{2}} + c$$
  • $$log \,\sec x (\sec x + \tan x) + c$$
  • None of these
$$\int(tanx -cotx)^{2}dx= $$
  • $$tanx+x+c$$
  • $$tanx-x+c$$
  • $$tanx-cotx+c$$
  • $$tanx-cotx-4x+cc$$
If $$\int \log \left( a ^ { 2 } + x ^ { 2 } \right) d x = h ( x )  ,$$ then $$h ( x ) = $$ 
  • $$x \log \left( a ^ { 2 } + x ^ { 2 } \right) + 2 \tan ^ { - 1 } \left( \dfrac { x } { a } \right) + c$$
  • $$x ^ { 2 } \log \left( a ^ { 2 } + x ^ { 2 } \right) + x + a \tan ^ { - 1 } \left( \dfrac { x } { a } \right) + c$$
  • $$x \log \left( a ^ { 2 } + x ^ { 2 } \right) - 2 x + 2 a \tan ^ { - 1 } \left( \dfrac { x } { a } \right) + c$$
  • $$x ^ { 2 } \log \left( a ^ { 2 } + x ^ { 2 } \right) + 2 x - a ^ { 2 } \tan ^ { - 1 } \left( \dfrac { x } { a } \right) + c$$
Evaluate: $$\displaystyle\int \frac { 1 } { ( x + 2 ) \sqrt { x + 1 } } d x $$
  • $$2 \tan ^ { - 1 } ( \sqrt { x + 1 } ) + c$$
  • $$\tan ^ { - 1 } ( \sqrt { x - 1 } ) + c$$
  • $$2 \tanh ^ { - 1 } ( \sqrt { x + 1 } ) - c$$
  • $$\tan ^ { - 1 } ( \sqrt { x + 1 } ) + c$$
Evaluate:
$$\int x ^ { x } \ln ( e ^x ) d x$$  
  • $$x ^ { x } + C$$
  • $$x \cdot \ln x + C$$
  • $$( \ln x ) ^ { x } + C$$
  • $$x ^ { \ln x } + C$$
 $$\displaystyle \int e^{x}[f(x)+f'(x)]dx=e^{x}.f(x)+C$$
  • True
  • False
$$\int \log _ { 10 } x d x =$$
  • $$x \log _ { 10 } x + c$$
  • $$x \left( \log _ { 10 } x + \log _ { 10 } e \right) + c$$
  • $$\log _ { 10 } x + c$$
  • $$x \left( \log _ { 10 } x - \log _ { 10 } e \right) + c$$
If $$\int { f\left( x \right)dx=f\left( x \right) } ,$$ then $$\int { \left\{ f\left( x \right) \right\}^2  }$$ dx is equal to :
  • $$\frac { 1 }{ 2 } \left\{ f\left( x \right) \right\}^2 $$
  • $${ \left\{ f\left( x \right) \right\}^3 } $$
  • $$\dfrac { { \left\{ f\left( x \right) \right\} ^{ 3 } } }{ 3 } $$
  • $${ \left\{ f\left( x \right) \right\}^2 } $$
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