CBSE Questions for Class 12 Commerce Applied Mathematics Indefinite Integrals Quiz 7 - MCQExams.com

Evaluate $$\displaystyle \int_{}^{} {x\sqrt {\frac{{{a^2} - {x^2}}}{{{a^2} + {x^2}}}} dx  } $$
  • $$\displaystyle \frac{1}{2}{a^2}{\cos ^{ - 1}}\left( {\frac{{{x^2}}}{{{a^2}}}} \right) + \frac{1}{2}\sqrt {{a^4} + {x^4}} + C$$
  • $$\displaystyle \frac{1}{2}{\sin ^{ - 1}}\left( {\frac{{{x^2}}}{{{a^2}}}} \right) + \sqrt {{a^4} + {x^4}} + C$$
  • $$\displaystyle \frac{1}{2}{a^2}{\sin ^{ - 1}}\left( {\frac{{{x^2}}}{{{a^2}}}} \right) + \frac{1}{2}\sqrt {{a^4} - {x^4}} + C$$
  • $$\displaystyle \frac{1}{2}{\cos ^{ - 1}}\left( {\frac{{{x^2}}}{{{a^2}}}} \right) + \frac{1}{2}\sqrt {{a^4} - {x^4}} + C$$
Solve:
$$\int 4x^2 e^{2x^3} \ dx$$
  • $$\dfrac{2}{3}e^{2x^3}+C$$
  • $$\dfrac{4}{3}e^{2x^3}+C$$
  • $$\dfrac{3}{2}e^{2x^3}+C$$
  • $$\dfrac{1}{3}e^{2x^3}+C$$

The value of $$\displaystyle\int {\dfrac{{\ln n\left( {1 - \left(
{\dfrac{1}{x}} \right)} \right)dx}}{{x\left( {x - 1} \right)}}} $$ is

  • $$\dfrac{1}{2}\left[ {m{{\left( {1 - \frac{1}{x}} \right)}^2}} \right] + C$$
  • $$\dfrac{1}{2}{\left[ {m\left( {1 + \frac{1}{x}} \right)} \right]^2} + C$$
  • $$\dfrac{1}{2}\ell n\left( {x\left( {x - 1} \right)} \right) + C$$
  • $$\dfrac{1}{2}{\left[ {\ell n(x(x - 1))} \right]^2} + C$$
If $$I=\int e^x\frac{(1-x)^2}{(1+x^2)^2}dx $$ , then I =
  • $$ e^x\frac{1-x^2}{(1+x^2)}+C$$
  • $$ e^x\frac{1-x}{(1+x^2)}+C$$
  • $$ e^x\frac{1}{(1+x^2)}+C$$
  • $$ e^x\frac{1-2x}{(1+x^2)^2}+C$$
The value of $$\displaystyle\int {{e^{{{\tan }^{ - 1}}x}}} \left( {\dfrac{{1 + x + {x^2}}}{{1 + {x^2}}}} \right)dx$$ is equal to
  • $$x{e^{{{\tan }^{ - 1}}x}} + C$$
  • $${x^2}{e^{{{\tan }^{ - 1}}x}} + C$$
  • $$\dfrac{1}{x}{e^{{{\tan }^{ - 1}}x}} + C$$
  • $$x{e^{{{\cot }^{ - 1}}x}} + C$$
Evaluate $$\displaystyle\int{\dfrac{x}{\sqrt{{{x}^{2}}+2}}dx}$$
  • $$I=\sqrt{{{x}^{2}}-2}+C$$
  • $$I=\sqrt{{{x}^{2}}+2}+C$$
  • $$I=\sqrt{{{x}^{3}}+2}+C$$
  • $$I=\sqrt{{{x}^{3}}-2}+C$$
The function $$f(x)$$ satisfying the equation $$f^{2}(x)+4f'(x).f(x)+[f'(x)]^{2}=0$$ is-
  • $$f(x)=c.e^{(2-\sqrt {3}x)}$$
  • $$f(x)=c.e^{-(2-\sqrt {3}x)}$$
  • $$f(x)=c.e^{(\sqrt {3}-2)x}$$
  • $$f(x)=c.e^{-(2+\sqrt {3})x}$$
The value of $$\int {\left( {x - 1} \right){e^{ - x}}} $$ is equal to 
  • $$ - x{e^x} + C$$
  • $$x{e^x} + C$$
  • $$ - x{e^{ - x}} + C$$
  • $$x{e^{ - x}} + C$$
$$\int ( 1+3x+3x^2+4x^3+........)dx (|x| <1)$$-
  • $$(1+x)^{-1}+c$$
  • $$(1-x)^{-1}+c$$
  • $$(1+x)^{-2}+c$$
  • None of these
$$ \displaystyle \int { \dfrac { { 2 }^{ x } }{ \sqrt { 1-{ 4 }^{ x } }  } dx=K\sin ^{ -1 }{ \left( { 2 }^{ x } \right) +C }  }$$, then the value of $$K$$ is equal to
  • $$\ell n 2$$
  • $$\dfrac{1}{2}\ell 2$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{\ell n 2}$$
$$\int { \cos { \left( \log { x }  \right)  }  } dx=............\quad +\quad c$$
  • $$\dfrac { x }{ 2 } \left[ \cos { \left( \log { x } \right) } +\sin { \left( \log { x } \right) } \right] $$
  • $$\dfrac { x }{ 4 } \left[ \cos { \left( \log { x } \right) } +\sin { \left( \log { x } \right) } \right] $$
  • $$\dfrac { x }{ 2 } \left[ \cos { \left( \log { x } \right) } -\sin { \left( \log { x } \right) } \right] $$
  • $$\dfrac { x }{ 2 } \left[ \sin { \left( \log { x } \right) } +\cos { \left( \log { x } \right) } \right] $$
$$\int x\sin x\sec^{3}xdx=$$
  • $$\dfrac {1}{2}[\sec^{2}x-\tan x]+c$$
  • $$\dfrac {1}{2}[x\sec^{2}x-\tan x]+c$$
  • $$\dfrac {1}{2}[x\sec^{2}x+\tan x]+c$$
  • $$\dfrac {1}{2}[\sec^{2}x+\tan x]+c$$
$$ \int { P\left( x \right) { e }^{ kx }dx=Q\left( x \right) { e }^{ 4x }+C }$$, where $$P(x)$$ is polynomial of degree $$n$$ and $$Q(x)$$ is polynomial of degree $$7$$. Then the value of $$ n+7+k+\lim _{ x\rightarrow \infty  }{ \dfrac { P\left( x \right)  }{ Q\left( x \right)  }  }$$ is:
  • $$18$$
  • $$19$$
  • $$20$$
  • $$22$$
Evaluate using limit of sum:
$$\displaystyle \int_{1}^{3} {(x+1)^2}dx$$
  • $$26$$
  • $$28$$
  • $$30$$
  • $$32$$
$$\int _{ 0 }^{ \pi /4 }{ \cfrac { \sec ^{ 2 }{ x }  }{ \left( 1+\tan { x }  \right) \left( 2+\tan { x }  \right)  } dx } $$ equals:
  • $$\log _{ e }{ \cfrac { 2 }{ 3 } } $$
  • $$\log _{ e }{ 3 } $$
  • $$\cfrac { 1 }{ 2 } \log _{ e }{ \cfrac { 4 }{ 3 } } $$
  • $$\log _{ e }{ \cfrac { 4 }{ 3 } } $$
If $$l_{n}=\displaystyle \int{\dfrac{t^{n}}{1+t^{2}}}dt$$ then
  • $$l_{n+2}=\dfrac{t^{n}}{n}-nl_{n}$$
  • $$l_{n+1}=\dfrac{t^{n+1}}{n+1}l_{n}$$
  • $$l_{n+1}=\dfrac{t^{n-1}}{n-1}l_{n}$$
  • $$l_{n21}=\dfrac{t^{n+1}}{n+1}l_{n}$$
$$\int {{e^x}\left[ {{\mathop{\rm tanx}\nolimits}  - log\left( {\cos x} \right)} \right]} dx = $$
  • $${e^x}\log \left( {\sec x} \right) + c$$
  • $${e^x}\log \left( {co\sec x} \right) + c$$
  • $${e^x}\log \left( {\cos x} \right) + c$$
  • $${e^x}\log \left( {\sin x} \right) + c$$
If $$\displaystyle \int \dfrac{dx}{\sqrt{\sin^3 x \cos^5 x}} = a \sqrt{\cot x } + b \sqrt {\tan^3x} + c$$ where c is an arbitrary constant of integration then the values of $$'a'$$ and $$'b'$$ are respectively :
  • $$-2 $$ & $$\dfrac{2}{3}$$
  • $$2 $$ & $$-\dfrac{2}{3}$$
  • $$2 $$ & $$\dfrac{2}{3}$$
  • None of these
$$\displaystyle \int (x^2-x+5)\, dx$$
  • $$\dfrac {x^3}3-\dfrac{x^2}2+5x+c$$
  • $$\dfrac {x^3}3+\dfrac{x^2}2+5x+c$$
  • $$\dfrac {x^2}2-\dfrac{x}2+5x+c$$
  • $$\dfrac {x^4}4-\dfrac{x^4}3+5$$
$$\displaystyle \int {{x^3}{e^{{x^2}}}dx = } $$
  • $$\dfrac{1}{2}\left( {{x^2} + 1} \right){e^{{x^2}}} + c$$
  • $$\left( {{x^2} + 1} \right){e^{{x^2}}} + c$$
  • $$\dfrac{1}{2}\left( {{x^2} - 1} \right){e^{{x^2}}} + c$$
  • $$\left( {{x^2} - 1} \right){e^{{x^2}}} + c$$
$$\displaystyle\int x^2e^{x^3}\cos \left(e^{x^3}\right)dx$$ is equal to?
  • $$\sin\left(e^{x^3}\right)+C$$
  • $$3\sin \left(e^{x^3}\right)+C$$
  • $$\dfrac{1}{3}\sin\left(e^{x^3}\right)+C$$
  • $$e^x\sin\left(e^{x^3}\right)+C$$
The value of $$\int  \sqrt { \dfrac { e^ x\quad -\quad 1 }{ e^ x\quad +\quad 1 }  } dx$$ is equal to
  • $$\ell n\left( e^ x+\sqrt { e^ 2x\quad -1 } \right) - sec^ -1(e^ x)+c$$
  • $$\ell n\left( e^ x+\sqrt { e^ 2x\quad -1 } \right) +sec^ -1(e^ x)+c$$
  • $$\ell n\left( e^ x-\sqrt { e^ 2x\quad -1 } \right) -sec^ -1(e^ x)+c$$
  • $$\ell n\left( e^ x+\sqrt { e^ 2x\quad -1 } \right) -sin^ -1(e^ {-x})+c$$
$$\int \sqrt {1 + \sin x}dx =$$
  • $$\dfrac {1}{2}\left (\sin \dfrac {x}{2} + \cos \dfrac {x}{2}\right ) + c$$
  • $$\dfrac {1}{2}\left (\sin \dfrac {x}{2} - \cos \dfrac {x}{2}\right ) + c$$
  • $$2\sqrt {1 + \sin x} + c$$
  • $$-2\sqrt {1 - \sin x} + c$$
The integral of $$\displaystyle\int e^{\sin x}(x\cos x-\sec x\tan x)dx$$ is?
  • $$se^{\sin x}-e^{\sin x}\sec x+c$$
  • $$(x+\sec x)e^{\sin x}+c$$
  • $$e^{\sin x}\cos x+c$$
  • $$e^{\sin x}(\cos x-\sec x)+c$$
The value of $$\int (x-1) e^{-x}$$ dx is equal to
  • $$-xe^x + C$$
  • $$xe^x + C$$
  • $$-xe^{-x} + C$$
  • $$xe^{-x} + C$$
$$\int \dfrac {dx}{(x^{2} + 4x + 5)^{2}}$$ is equal to
  • $$\dfrac {1}{2}\left [\tan^{-1}(x + 1) + \dfrac {x + 2}{x^{2} + 4x + 5}\right ] + c$$
  • $$\dfrac {1}{2}\left [\tan^{-1}(x + 2) - \dfrac {x + 2}{x^{2} + 4x + 5}\right ] + c$$
  • $$\dfrac {1}{2}\left [\tan^{-1}(x + 1) - \dfrac {x + 2}{x^{2} + 4x + 5}\right ] + c$$
  • $$\dfrac {1}{2}\left [\tan^{-1}(x + 2) + \dfrac {x + 2}{x^{2} + 4x + 5}\right ] + c$$
$$\int\dfrac{e^x+e^{-x}+(e^x-e^{-x})sin x}{1+cos x}dx=$$
  • $$(e^x+e^{-x}) tan (x/2)+C$$
  • $$(e^x-e^{-x}) cot (x/2)+C$$
  • $$(e^x-e^{-x}) tan (x/2)+C$$
  • $$(e^x-e^{-x}) cosec (x/2)+C$$
If $$\displaystyle \int \dfrac{x^4 + 1}{x^6 + 1} dx = \tan^{-1} (f(x)) -\dfrac{2}{3} \tan^{-1} (g(x)) + C$$, then
  • Both $$f(x)$$ & $$g(x)$$ are odd functions
  • $$g(x)$$ is monotonic function
  • none of these
  • None 
Integral of $$f ( x ) = \sqrt { 1 + x ^ { 2 } }$$ with respect to $$x ^ { 2 }$$ is
  • $$\frac { 2 } { 3 } \frac { \left( 1 + x ^ { 2 } \right) ^ { 3 / 2 } } { x } + k$$
  • $$\frac { 2 } { 3 } \left( 1 + x ^ { 2 } \right) ^ { 3 / 2 } + k$$
  • $$\frac { 2 } { 3 } x \left( 1 + x ^ { 2 } \right) ^ { 3 / 2 } + k$$
  • None of these
 The value of $$\displaystyle \int {\dfrac{{dx}}{{\sin x.\sin \left( {x + \alpha } \right)}}} $$ equal to
  • $$\csc\,\,\alpha \,\,\ell n\left| {\dfrac{{\sin x}}{{\sin \left( {x + \alpha } \right)}}} \right| + c$$
  • $$\;\;\;\;\csc\,\,\alpha \,\,\ell n\left| {\dfrac{{\sin \left( {x + \alpha } \right)}}{{\sin x}}} \right| + c$$
  • $$\csc\,\,\alpha \,\,\ell n\left| {\dfrac{{\sec \left( {x + \alpha } \right)}}{{\sec x}}} \right| + c\;$$
  • $$\,\;\;\csc\,\,\alpha \,\,\ell n\left| {\dfrac{{\sec x}}{{\sec \left( {x + \alpha } \right)}}} \right| + c$$
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