CBSE Questions for Class 12 Commerce Applied Mathematics Indefinite Integrals Quiz 1 - MCQExams.com

$$\displaystyle \int_{0}^{1}\tan ^{-1}x\:dx.$$
  • $$\displaystyle \frac{\pi }{2}-\frac{1}{2}\log 2.$$
  • $$\displaystyle \frac{\pi }{4}+\frac{1}{2}\log 2.$$
  • $$\displaystyle \frac{\pi }{2}-\frac{1}{4}\log 2.$$
  • $$\displaystyle \frac{\pi }{4}-\frac{1}{2}\log 2.$$
$$\int \dfrac {dx}{\sqrt {x^{10} - x^{2}}}; x > 1=$$ ______ $$+ C$$.
  • $$\dfrac {1}{4}\log |\sqrt {x^{10} - x^{2}} + x^{2}|$$
  • $$\dfrac {1}{2}\log |x^{10} - x^{2}|$$
  • $$-\dfrac {1}{4}\sec^{-1} (x^{4})$$
  • $$\dfrac {1}{4}\sec^{-1} (x^{4})$$
The value of $$\int  log  x  dx$$ is
  • $$\displaystyle x log \left ( \frac{x}{e} \right ) + c$$
  • $$x^2 log x + c$$
  • $$e^x log x + c$$
  • $$(x + 1) log x + c$$
$$\displaystyle \int x.e^{x}dx=$$
  • $$ x.e^{x}+c$$
  • $$ e^{x}(x-1)+c$$
  • $$ e^{x}(x+1)+c$$
  • $$ e^{x}(x+2)+c$$
$$\displaystyle \int x^{3}e^{x}dx=$$
  • $$e^{x}(x^{3}-3x^{2}+6x+6)+c$$
  • $$e^{x}(x^{3}- 3x^{2}+6x-6)+c$$
  • $$e^{x}(x^{3}+3x^{2}+6x+6)+c$$
  • $$e^{x}(x^{3}- 3x^{2}-6x+6)+c$$
$$\displaystyle \int\log\sqrt{x+1}dx=$$
  • $$\displaystyle \frac{1}{2}[(x+1)\log (x+1)-x]+c$$
  • $$\displaystyle \frac{1}{2}[(x-1)\log(x+1)-x]+c$$
  • $$\displaystyle \frac{1}{2}[x \displaystyle \log(x+1)-\frac{x^{2}}{2}]+c$$
  • $$\displaystyle \frac{1}{2}[(x+1)\displaystyle \log(x+1)-\frac{x}{2}]+c$$
$$\displaystyle \int[\sin(\log x)+\cos(\log x)]dx=$$
  • $$\displaystyle e^{x} \sin (\log x)+c$$
  • $$\displaystyle e^{x} \cos (\log x)+c$$
  • $$\displaystyle x\sin (\log x)+c$$
  • $$\displaystyle x\cos (\log x)+c$$
The integral $$\displaystyle  \int (1+x-\displaystyle \frac{1}{x})e^{x+\frac{1}{x}} dx $$ is equal to 
  • $$ (x-1)e^{x+ \frac{1}{x}} +c $$
  • $$ xe^{x+\frac{1}{x}} +c $$
  • $$ (x+1)e^{x+\frac{1}{x}} +c $$
  • $$ -xe^{x+\frac{1}{x}} +c $$
The integral $$\displaystyle  \int x { \cos^{ -1 }\left(\displaystyle  \frac { 1-x^{ 2 } }{ 1+x^{ 2 } }  \right) dx } $$ is equal to :
(Note : $$(x>0)$$)
  • $$-x + (1+x^2) \cot^{-1} x+c$$
  • $$-x+(1+x^{2})\tan^{-1}x+c$$
  • $$-x-(1+x^{2})\tan^{-1}xc$$
  • $$x-(1+x^2)\cot^{-1}x+c$$
If $$\int f(x)\ dx $$ $$=\Psi(x)$$ , then $$\int x^{5}f(x^{3})\ dx $$ is equal to
  • $$\dfrac{1}{3}x^{3}\Psi(x^{3})-3\int x^{3}\Psi(x^{3})dx+C$$
  • $$\dfrac{1}{3}x^{3}\Psi(x^{3})-\int x^{2}\Psi(x^{3})dx+C$$
  • $$\dfrac{1}{3}[x^{3}\Psi(x^{3})-\int x^{3}\Psi(x^{3}) dx ]+C$$
  • $$\dfrac{1}{3}[x^{3}\Psi(x^{3})-\int x^{2}\Psi(x^{3}) dx ]+C$$
If $$f''(x)=-f(x)$$ and $$g(x)=f'(x)$$ and $$\displaystyle{F}({x})=\left ({f}\left (\frac{{x}}{2}\right)\right)^{2}+\left ({g}\left (\frac{{x}}{2}\right)\right)^{2}$$ and given that $$F(5)=5$$, then $$F(10)$$ is equal to 
  • $$5$$
  • $$10$$
  • $$0$$
  • $$15$$
$$\int \frac{ f(x) g'(x)-f'(x)g(x)}{f(x)g(x)}\begin{bmatrix} Log (g(x))-Log (f(x)) \end{bmatrix}dx=$$
  • $$Log\begin{pmatrix} \frac {g(x)}{f(x)}\end{pmatrix}+C$$
  • $$\frac {1 }{2 } \begin {vmatrix} Log \begin{pmatrix} \frac {g(x)}{f(x)}\end{pmatrix} \end {vmatrix}^2+C$$
  • $$\dfrac {(g)x}{f(x)}Log \begin {pmatrix}\frac {g(x)}{f(x)}\end {pmatrix} +C$$
  • $$Log\begin{bmatrix} \frac {g(x)}{f(x)} \end{bmatrix}-\frac {g(x)}{f(x)}+C$$
$$\int x  log  x  dx $$ is equal to
  • $$\displaystyle\frac{x^2}{4}(2log x-1)+c$$
  • $$\displaystyle\frac{x^2}{2}(2log x-1)+c$$
  • $$\displaystyle\frac{x^2}{4}(2log x+1)+c$$
  • $$\displaystyle\frac{x^2}{2}(2log x+1)$$
The value of $$\int { { e }^{ \tan { \theta  }  } } \left( \sec { \theta  } -\sin { \theta  }  \right) d\theta$$ is equal to ?
  • $$-{ e }^{ \tan { \theta } }\sin { \theta } +C$$
  • $${ e }^{ \tan { \theta } }\sin { \theta } +C$$
  • $${ e }^{ \tan { \theta } }\sec { \theta } +C$$
  • $${ e }^{ \tan { \theta } }\cos { \theta } +C$$
$$\displaystyle \int { \log { \left( \log { x }  \right) +\dfrac { 1 }{ \log { x }  }  }  } dx$$
  • $$x\log { \left( \log { x } \right) +c }$$
  • $$x\log { x } +c$$
  • $$x+c$$
  • $$None$$
The integral of $$\displaystyle\int e^{x} (\sin x+\cos x)dx$$ is
  • $$e^{ x} \cos x + c$$
  • $$e^{x}\sin x + c$$
  • $$e^{ x}\sec x + c$$
  • none of this
If $$\displaystyle\int { \frac { { 2 }^{ x } }{ \sqrt { 1-{ 4 }^{ x } }  }  } dx=K\sin ^{ -1 }{ \left( { 2 }^{ x } \right)  } +C$$, then the value of $$K$$ is equal to
  • $$\ell n\ 2$$
  • $$\dfrac {1}{2} \ell n\ 2$$
  • $$\dfrac {1}{2}$$
  • $$\dfrac {1}{\ell n\ 2}$$
The value of integral $$\int {{{\tan }^{ - 1}}\left( {\dfrac{{{x^3}}}{{1 + {x^2}}}}
\right)}  + {\tan ^{ - 1}}\left({\dfrac{{1 + {x^2}}}{{{x^3}}}} \right)dx$$ is equal to 
  • $$1$$
  • $$ - \dfrac{\pi }{2} + c $$
  • $$\dfrac{\pi }{2} + c $$
  • $$\left( {\dfrac{\pi }{2}} \right)x + c $$

$$ \int ( e^{\log x} + \sin x) \cos x \  dx$$ is equal to
  • $$ x \sin x + \cos x - \dfrac {\sin^2x}{2}+ c $$
  • $$ x \cos x - \sin^2x+ c $$
  • $$ x \sin x + \cos x - (\cos^2x)/2+ c $$
  • $$ x^2 \sin^x + \cos x - \sin^3 x + c $$
$$\int_{}^{} {x{{\sin }^{ - 1}}xdx} $$
  • $$\frac{1}{4}{\sin ^{ - 1}}x\left( {2x + 1} \right) + \frac{{x\sqrt {1 - {x^2}} }}{4} + c$$
  • $$\frac{1}{4}{\sin ^{ - 1}}x\left( {2{x^2} + 1} \right) + \frac{{x\sqrt {1 - {x^2}} }}{4} + c]$$
  • $$\frac{1}{4}{\cos ^{ - 1}}x\left( {2x + 1} \right) + \frac{{x\sqrt {1 - {x^2}} }}{4} + c$$
  • $$\frac{1}{4}{\sin ^{ - 1}}x\left( {2x + 1} \right) - \frac{{x\sqrt {1 - {x^2}} }}{4} + c$$
Solve $$\int {\dfrac{1}{{\sqrt[{}]{{9 - 25{x^2}}}}}} dx$$
  • $$ \dfrac{1}{5}{{\sin }^{-1}}\left( \dfrac{5x}{3} \right)+C $$
  • $$ {{\sin }^{-1}}\left( \dfrac{5x}{3} \right)+C $$
  • $$ \dfrac{1}{5}{{\sin }^{-1}}\left( \dfrac{3x}{5} \right)+C $$
  • $${{\sin }^{-1}}\left( \dfrac{3x}{5} \right)+C $$
Find the general solution of $$\frac{{dy}}{{dx}} = \frac{{2y}}{x}$$
  • $$ y=e^{5\log x +c}$$
  • $$ y=e^{3\log x +c}$$
  • $$ y=e^{2\log x +c}$$
  • None of these
The value of $$\int _{  }^{  }{ \cfrac { \log { x }  }{ { \left( x+1 \right)  }^{ 2 } }  } dx$$ is
  • $$\cfrac { -\log { x } }{ x+1 } +\log { x } -\log { \left( x+1 \right) } +C $$
  • $$\cfrac { \log { x } }{ x+1 } +\log { x } -\log { \left( x+1 \right) } +C  $$
  • $$\cfrac { \log { x } }{ x+1 } -\log { x } -\log { \left( x+1 \right) } + C $$
  • $$\cfrac { -\log { x } }{ x+1 } -\log { x } -\log { \left( x+1 \right) } +C  $$
$$\displaystyle \int \sin x \log (\sec x+\tan x)dx=f(x)+x+c$$ then $$f(x)=$$
  • $$\cos x \log(\sec x + \tan x)+c$$
  • $$\sin x\log(\sec x+\tan x)+c$$
  • $$- \cos x \log\sec x+\tan x)+c$$
  • $$-\cos x\log \sec x+c$$
$$\int [\frac{f(x)g'(x)-f'(x)g(x)}{f(x)g(x)}].[log (g(x))-log(f(x))]dx=$$
  • $$log\frac{g(x)}{f(x)}+c$$
  • $$\frac{1}{2}(log\frac{g(x)}{f(x)})^{2}+c$$
  • $$\frac{g(x)}{f(x)}log(\frac{g(x)}{f(x)})+c$$
  • $$\frac{f(x)}{g(x)}log(\frac{g(x)}{f(x)})+c$$
$$\frac{1}{2}\int \sqrt{x^{3}-8x^{2}dx}$$ equals
  • $${\frac{(x-8)^{5/2}}{5}+\frac{8(x-8)^{3/2}}{3}}+c$$
  • $${\frac{(x-8)^{5/2}}{5}+\frac{8}{3}(x-8)^{3/2}}+c$$
  • $${\frac{(x-8)^{5/2}}{3}+\frac{8}{5}(x-8)^{3/2}}+c$$
  • None of these
$$\int\ \sec\theta\ d\theta$$
  • $$\dfrac{(\sec\theta+\tan\theta)}{2}+c$$
  • $$\dfrac{(\sec\theta+\tan\theta)}{3}[2+4\tan\theta(\sec\theta+\tan\theta)]+c$$
  • $$\ln {(|\sec\theta+\tan\theta|)}+c$$
  • $$\dfrac{3(\sec\theta+\tan\theta)}{2}[2+\tan\theta(\sec\theta+\tan\theta)]+c$$
$$\int {\dfrac{{\cot \sqrt x }}{{2\sqrt x }}dx} $$ is equal to $$ = \_\_\_\_\_ + C.$$
  • $$2\log |\sin \sqrt x |$$
  • $$\log |\sin \sqrt x |$$
  • $$\dfrac{1}{2}\log |\sin \sqrt x |$$
  • None of these
Evaluate :
$$\int \cos^3 x e^{\log \sin x} dx$$
  • $$-\dfrac{\cos ^4x}{4}+C$$
  • $$\dfrac{\sin x}{x^2}+C$$
  • $$-\dfrac{\sin^3x}{3}+C$$
  • None of these
The value of $$\int { (x-1) } { e }^{ -x }$$ dx is equal to 
  • $$-{ xe }^{ x }+C$$
  • $${ xe }^{ x }+C$$
  • $$-{ xe }^{ -x }+C$$
  • $${ xe }^{ -x }+C$$
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