CBSE Questions for Class 12 Commerce Applied Mathematics Indefinite Integrals Quiz 6 - MCQExams.com

$$\int e^x (\sin  x + 2 \cos  x) \sin x\,  dx$$ is equal to
  • $$e^x \cos x + C$$
  • $$e^x \sin x + C$$
  • $$e^x \sin^2 x + C$$
  • $$e^x \sin 2x + C$$
  • $$e^x (\cos x + \sin x) + C$$
$$\displaystyle\int { \cfrac { 1 }{ 7 } \sin { \left( \cfrac { x }{ 7 } +10 \right)  } dx } $$ is equal to
  • $$\cfrac { 1 }{ 7 } \cos { \left( \cfrac { x }{ 7 } +10 \right) } +C$$
  • $$-\cfrac { 1 }{ 7 } \cos { \left( \cfrac { x }{ 7 } +10 \right) } +C$$
  • $$-\cos { \left( \cfrac { x }{ 7 } +10 \right) } +C$$
  • $$-7\cos { \left( \cfrac { x }{ 7 } +10 \right) } +C$$
  • $$\cos { \left( x+70 \right) } +C$$
$$\displaystyle \int \dfrac{xe^x}{(1 + x)^2} dx$$ is equal to
  • $$\dfrac{-e^x}{x + 1} + C$$
  • $$\dfrac{e^x}{x + 1} + C$$
  • $$\dfrac{xe^x}{x + 1} + C$$
  • $$\dfrac{-xe^x}{x + 1} + C$$
  • $$\dfrac{e^x}{(x + 1)^2} + C$$
If $$\displaystyle\int { \sqrt { 1+\sin { x }  } \cdot f\left( x \right) dx } =\dfrac { 2 }{ 3 } { \left( 1+\sin { x }  \right)  }^{ { 3 }/{ 2 } }+C$$, then $$f\left( x \right) $$ is equal to
  • $$\cos { x } $$
  • $$\sin { x } $$
  • $$\tan { x } $$
  • $$1$$
$$\displaystyle\int { { x }^{ x }\log { \left( ex \right)  } dx } $$ is equal to
  • $${ x }^{ x }+c$$
  • $$x\cdot \log { x } +c$$
  • $${ \left( \log { x } \right) }^{ x }+c$$
  • $${ x }^{ \log { x } }+c$$
$$\int \dfrac {2x + 5}{\sqrt {7 - 6x - x^{2}}} dx = A\sqrt {7 - 6x - x^{2}} + B\sin^{-1} \left (\dfrac {x + 3}{4}\right ) + C$$
(where $$C$$ is a constant of integration), then the ordered pair $$(A, B)$$ is equal to
  • $$(-2, -1)$$
  • $$(2, -1)$$
  • $$(-2, 1)$$
  • $$(2, 1)$$
If $$f\left( x \right)$$ satisfies the relation $$f\left( \dfrac { 5x-3y }{ 2 }  \right) =\dfrac { 5f\left( x \right) -3f\left( y \right)  }{ 2 } \forall \ x,y\in R$$ and $$\ f\left( 0 \right) =3$$ and $$f^{ ' }\left( 0 \right) =2$$, then the period of $$\sin { \left( f\left( x \right)  \right)  }$$ is
  • $$2\pi$$
  • $$\pi$$
  • $$3\pi$$
  • $$4\pi$$
$$\displaystyle\int\displaystyle\frac{e^{\cot^{-1}x}}{1+x^2}(x^2-x+1)dx$$ 
  • $$\displaystyle\frac{e^{\cot^{-1}x}}{1+x^2}$$
  • $$x\cdot e^{\cot^{-1}x}$$
  • $$e^{\cot^{-1}x}$$
  • $$-e^{\cot^{-1}x}$$
Let $$I=\displaystyle \int _{ \pi /4 }^{ \pi /3 }{ \cfrac { \sin { x }  }{ x }  } dx$$. Then?
  • $$\cfrac { 1 }{ 2 } \le I\le 1\quad $$
  • $$4\le I\le 2\sqrt { 30 } $$
  • $$\cfrac { \sqrt { 3 } }{ 8 } \le I\le \cfrac { \sqrt { 2 } }{ 6 } $$
  • $$1\le I\le \cfrac { 2\sqrt { 3 } }{ \sqrt { 2 } } $$
$$\int { { \left( ex \right)  }^{ x }\left( 2+\log { x }  \right)  } dx=....+c,x\in { R }^{ + }-\left\{ 1 \right\} $$
  • $${ x }^{ x }$$
  • $${ \left( ex \right) }^{ x }\quad $$
  • $${ e }^{ x }$$
  • $$\left( 1+\log { x } \right) { \left( ex \right) }^{ x }$$
$$\displaystyle \int \tan^{-1} x dx=$$__________$$+c$$.
  • $$x\tan^{-1}x+\dfrac{1}{2}log\left|\dfrac{\tan^{-1}x}{x^2+1}\right|$$
  • $$x\tan^{-1}x-\dfrac{1}{2}log\left|x^2+1\right|$$
  • $$x\tan^{-1}x+\dfrac{1}{2}log\left|x^2+1\right|$$
  • $$\dfrac{1}{1+x^2}$$
Solve  $$\int { { e }^{ 7x+\log { x }  } } dx=.....+c$$
  • $${ e }^{ 7x }\left[ 7x-1 \right] $$
  • $$\cfrac { { e }^{ x } }{ 98 } \left[ 7x-1 \right] \quad $$
  • $$\cfrac { { e }^{ x } }{ 49 } \left[ 7x-1 \right] \quad $$
  • $${ e }^{ 7x }\quad $$
If $$\displaystyle\int f(x)dx=g(x),$$ then $$\displaystyle\int f^{-1}(x)dx=$$ _____________$$+$$c.
  • $$x\cdot f(x)-g(f^{-1}(x))$$
  • $$xf^{-1}(x)-g(f^{-1}(x))$$
  • $$xf^{-1}(x)-g(f(x))$$
  • $$x\cdot f^{-1}(x)$$
The value of $$\int \dfrac{d^2}{dx^2} \, (tan^{-1}x) \, dx$$ is equal to
  • $$\dfrac{1}{1 \, + \, x^2}$$ + C
  • $$tan^{-1}x$$ + C
  • x tan - $$\dfrac{1}{2} \, log (1 \, + \, x^2) \, + \, C$$
  • $$\dfrac{1 \, + \, x^2}{2}$$ + C
$$\int {\left( 3.{ x }^{ 2 }.\tan ^{ -1 }{ x } +\cfrac { { x }^{ 3 } }{ 1+{ x }^{ 2 } }  \right) } dx=....+c$$.
  • $${ x }^{ 3 }\tan ^{ -1 }{ x } $$
  • $$\cfrac { { x }^{ 3 } }{ 3 } \tan ^{ -1 }{ x } $$
  • $${ x }^{ 2 }\tan ^{ -1 }{ x } $$
  • $$\cfrac { { x }^{ 2 } }{ 2 } \tan ^{ -1 }{ x } $$
$$I= \int \frac{x+2}{(x+1)^2}dx;$$ then I is equal to 
  • $$\log (x+1)+\dfrac{1}{x+1}+c$$
  • $$\log (x+2)-\dfrac{1}{x+1}+c$$
  • $$\log (1+x)-\dfrac{1}{x+1}+c$$
  • $$\log (x+2)+\dfrac{1}{x+1}+c$$
If $$g(1)=g(2)$$ then the value of  $$\int _{ 1 }^{ 2 }{ { \left[ f\{ g(x)\}  \right]  }^{ -1 } } f'\{ g(x)\} g'(x)dx\quad is$$
  • $$1$$
  • $$2$$
  • $$0$$
  • none of these
$$\int {{2^x}.{e^x}dx = } $$
  • $${{{2^x}.{e^x}} \over {1 - \ell n2}} + C$$
  • $${{{2^x}.{e^x}} \over {1 + \ell n2}} + C$$
  • $${{{2^x}.{e^x}} \over { - 1 + \ell n2}} + C$$
  • $${{{{\left( {2e} \right)}^x}} \over {\ell n\left( {2e} \right)}} + C$$
$$y= \int \sqrt{1+\sin2x} dx; y $$ is equal to-
  • $$\sin x - \cos x+c$$
  • $$\sin x+ \cos x +c$$
  • $$ \cos x - \sin x +c$$
  • None of these
If $$I^n \, = \,  {I_{0}}^{\pi/4} \, tan ^n \, X   sec ^2 X dx $$     then $$I_1 , \, I _2 \, , \, I_3 \, $$ are in
  • A. P
  • G . P
  • H . M
  • none
If $$\displaystyle \int _0^{\pi/2} \sin x \cos x dx $$ is equal to:
  • $$\dfrac 1 2$$
  • $$\dfrac 14$$
  • $$2$$
  • $$1$$
The value of $$\displaystyle \int \dfrac{dx}{4x^2 +9}$$ is:
  • $$\dfrac{1}{6} \tan^{-1} \left(\dfrac{3x}{2}\right) + c$$
  • $$\dfrac{1}{6} \tan^{-1} \left(\dfrac{2x}{3}\right) + c$$
  • $$\dfrac{3}{2} \tan^{-1} \left(\dfrac{2x}{3}\right) + c$$
  • $$\dfrac{1}{2} \tan^{-1} \left(\dfrac{2x}{3}\right) + c$$
Evaluate : $$\displaystyle \int \sqrt{\dfrac{a - x}{x}} dx$$
  • $$\sqrt{a-x} \sqrt{x}+a\tan^{-1}\left(\dfrac{\sqrt{x}}{\sqrt{a-x}}\right)+c$$
  • $$\dfrac{\sqrt{a-x}}{\sqrt{x}}+a\tan^{-1}\left(\dfrac{\sqrt{x}}{\sqrt{a-x}}\right)+c$$
  • $$\dfrac{\sqrt{a-x}}{\sqrt{x}}-a\tan^{-1}\left(\dfrac{\sqrt{x}}{\sqrt{a-x}}\right)+c$$
  • $$\sqrt{a-x} \sqrt{x}-a\tan^{-1}\left(\dfrac{\sqrt{x}}{\sqrt{a-x}}\right)+c$$
Find the derivative of $$\dfrac{e^{x}}{\sin x}$$.
  • $$e^x\text{cosec }x[\cot x+1]$$
  • $$e^x\text{cosec }x[1-\cot x]$$
  • $$e^x\sec x[\cot x+1]$$
  • $$e^x\sec x[\cot x-1]$$
If $${ I }_{ n }=\int { \sin ^{ n }{ x } dx },$$, then $${ nI }_{ n }-\left( n-1 \right) { I }_{ n-2 }=$$
  • $$\sin ^{ n-1 }{ x\cos { x } }$$
  • $$\cos ^{ n-1 }{ x\sin{ x } }$$
  • $$-\sin ^{ n-1 }{ x\cos { x } }$$
  • $$-\cos ^{ n-1 }{ x\sin{ x } }$$
Solve $$\displaystyle \int {\frac{{x{e^x}}}{{{{\left( {x + 1} \right)}^2}}}dx} $$
  • $$\dfrac {e^x}{x+1}+C$$
  • $$\dfrac {x}{(x+1)^2}+C$$
  • $$e^x(x+1)+C$$
  • $$x(x+1)^2+C$$
The value of $$\int { { e }^{ { \tan ^{ -1 } { x }  } }} \dfrac { \left( 1+x{ +x }^{ 2 } \right)  }{ 1{ +x }^{ 2 } } dx$$ is ?
  • $$\tan ^{ -1 }{ x } +C$$
  • $$e^{\tan^{-2}{x}}+2x+C$$
  • $$e^{\tan^{-1}{x}}+C$$
  • $${ xe }^{ \tan ^{ -1 }{ x+C } }$$
If $$\displaystyle \int {\dfrac{{{2^x}}}{{\sqrt {1 - {4^x}} }}dx = K{{\sin }^{ - 1}}\left( {{2^x}} \right) + C} ,$$ then K is equal to 
  • $$\ell n2$$
  • $$\dfrac{1}{2}\ell n2$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{{\ell n2}}$$
$$\displaystyle \int \sqrt{\dfrac{a+x}{a-x}}dx$$ is equal to-
  • $$a\sin^{-1} (x/a)-\sqrt{a^{2}-x^{2}}+c$$
  • $$a\cos^{-1} (x/a)-\sqrt{a^{2}-x^{2}}+c$$
  • $$a\sin^{-1} (x/a)-\sqrt{a^{2}+x^{2}}+c$$
  • $$a\cos^{-1} (x/a)-\sqrt{a^{2}+x^{2}}+c$$
Evaluate: $$\displaystyle\int \dfrac{\sin^3x}{( \cos^4x+ 3 \cos^2x+ 1)\tan^{-1} ( \sec x+ \cos x)}dx$$
  • $$\tan^{-1} ( \sec x + \cos x) +c$$
  • $$\log_e | \tan^{-1} ( \sec x + \cos x) |+c$$
  • $$\dfrac{1}{(\sec x+ \cos x) ^2}+c$$
  • $$\tan^{-1} ( \cot x+ \text{cosec} x)+c$$
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