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

π011+sinx dx is equal to
  • 1
  • 2
  • 3
  • 4
If \quad f(x)=\begin{cases} 2{ x }^{ 2 }+1,x\le 1 \\ 4{ x }^{ 2 }-1,x>1 \end{cases}, then \int _{ 0 }^{ 2 }{ f(x)dx } is
  • 10
  • 50/3
  • 1/3
  • 47/2
What is \displaystyle \int_0^1 {\frac{\tan^{-1}x}{1 + x^2} dx} equal to ?
  • \dfrac{\pi}{4}
  • \dfrac{\pi}{8}
  • \dfrac{\pi^2}{8}
  • \dfrac{\pi^2}{32}
If I_n = \int_0^{\pi/4} tan^n \theta d \theta, where n is a positive integer, then n(I_{n-1} + I_{n +1}) is equal to
  • 1
  • n - 1
  • \dfrac{1}{n - 1}
  • None of these
The value of \displaystyle \int_{1/n}^{(an - 1)/n} \dfrac {\sqrt {x}}{\sqrt {a - x} + \sqrt {x}} dx is equal to
  • \dfrac {a}{2}
  • \dfrac {n\cdot a + 2}{2n}
  • \dfrac {n\cdot a - 2}{2n}
  • \dfrac {n\cdot a}{2}
The value of \displaystyle\int _{ 0 }^{ \pi  }{ \dfrac { dx }{ 5+3\cos { x }  }  } is
  • \dfrac{ \pi }{ 4 }
  • \dfrac{ \pi }{ 8 }
  • \dfrac{ \pi }{ 2 }
  • Zero
If \displaystyle \int_{\log 2}^{x}\frac{1}{\sqrt{e^x-1}}dx=\frac{\pi }{6}, then the value of x is 
  • \log 2
  • \log 3
  • \log 4
  • None of these
\displaystyle\int^{\pi /2}_{-\pi /2}\cos x ln \left(\displaystyle\frac{1+x}{1-x}\right)dx is equal to.
  • 0
  • \displaystyle\frac{\pi^2}{4}\left(-1+\frac{\pi}{2}\right)
  • 1
  • \displaystyle\frac{\pi^2}{2}
\displaystyle\int _{ 0 }^{ { \sqrt { \pi  }  }/{ 2 } }{ 2{ x }^{ 3 }\sin { \left( { x }^{ 2 } \right)  } dx } is equal to
  • \dfrac { 1 }{ \sqrt { 2 } } \left( 1+\dfrac { \pi }{ 4 } \right)
  • \dfrac { 1 }{ \sqrt { 2 } } \left( 1-\dfrac { \pi }{ 4 } \right)
  • \dfrac { 1 }{ \sqrt { 2 } } \left( \dfrac { \pi }{ 2 } -1 \right)
  • \dfrac { 1 }{ \sqrt { 2 } } \left( 1-\dfrac { \pi }{ 2 } \right)
  • \dfrac { 1 }{ \sqrt { 2 } } \left( \dfrac { \pi }{ 4 } -1 \right)
The value of \displaystyle\int _{ 0 }^{ { x }/{ 4 } }{ \dfrac { \sec { x }  }{ { \left( \sec { x } +\tan { x }  \right)  }^{ 2 } } dx } is
  • 1+\sqrt{2}
  • -11+\sqrt{2}
  • -\sqrt{2}
  • None of these
If f\left( x \right) is defined \left[ -2,2 \right] by f\left( x \right) =4{ x }^{ 2 }-3x+1 and g\left( x \right) =\dfrac { f\left( -x \right) -f\left( x \right)  }{ { x }^{ 2 }+3 } , then \displaystyle\int _{ -2 }^{ 2 }{ g\left( x \right) dx } is equal to
  • 64
  • -48
  • 0
  • 24
If \displaystyle \int_0^{\pi} \dfrac{x^2}{(1+\sin \,x)^2} dx = A the \displaystyle \int_0^{\pi} \dfrac{2x^2cos^2(x/2)}{(1+\sin x)^2} dx= ?
  • A+\pi - \pi^2
  • A-\pi+ \pi^2
  • A-\pi - \pi^2
  • A+2\pi - \pi^2
If I=\displaystyle\int _{ { 1 }/{ \pi  } }^{ \pi  }{ \dfrac { 1 }{ x } \cdot \sin { \left( x-\dfrac { 1 }{ x }  \right)  } dx } , then I is equal to
  • 0
  • \pi
  • \pi - \dfrac{1}{\pi}
  • \pi + \dfrac{1}{\pi}
Evaluate \displaystyle\int^4_1x\sqrt{x}dx
  • 12.8
  • 12.4
  • 7
  • None of these
The value of the definite integral \int_{1}^{\infty}(e^{x + 1} + e^{3 - x})^{-1}dx is
  • \dfrac {\pi}{4e^{2}}
  • \dfrac {\pi}{4e}
  • \dfrac {1}{e^{2}}\left (\dfrac {\pi}{2} - \tan^{-1}\dfrac {1}{e}\right )
  • \dfrac {\pi}{2e^{2}}
\displaystyle \int _{ 0 }^{ \pi/2}{ \frac{1}{a + b \cos x}} dx, a > |b| =.
  • \displaystyle \frac{2}{\sqrt{a^2 - b^2}} \tan^{-1} \sqrt{\frac{a + b}{a - b}}
  • \displaystyle \frac{2}{\sqrt{a^2 - b^2}} \cot^{-1} \sqrt{ \frac{a - b}{a + b} }
  • \displaystyle \frac{2}{\sqrt{a^2 - b^2}} \tan^{-1} \sqrt{ \frac{a - b}{a + b}}
  • \displaystyle \frac{\pi}{\sqrt{a^2 - b^2}}.
Solve \int_{0}^{\dfrac {\pi}{2}}\sqrt {\sin \phi}\cos^{5}\phi d\phi.
  • \dfrac{64}{231}
  • \dfrac{24}{231}
  • \dfrac{54}{231}
  • None of these
If \int _0^1 xdx = \dfrac {\pi}{4} - \dfrac {1}{2} ln 2 then the value of definite integral \int _0^1 \tan^{-1} (1-x+x^2) dx equals :
  • ln2
  • \dfrac {\pi}{4} + ln 2
  • \dfrac {\pi}{4} - ln2
  • 2 ln 2
\displaystyle \int_{0}^{\infty} \dfrac {x \ln x}{(1 + x^{2})^{2}}dx =
  • 1
  • -1
  • 0
  • \dfrac {\pi}{2}
Given \int_{1}^{2} e^{x^{2}} dx = a the the value of \int_{e}^{e^{4}}\sqrt {ln x} dx is
  • 2e^{4} - e - a
  • e^{4} - a
  • 2e^{4} - a
  • e^{4} - e
If \cfrac { \pi  }{ 2 } <t<\cfrac { 2\pi  }{ 3 } and =\int _{ 0 }^{ \sin { 2t }  }{ \cfrac { dx }{ \sqrt { 4\cos ^{ 2 }{ t-{ x }^{ 2 } }  }  }  } then the value of \cfrac { 2005\left( I+t \right)  }{ \pi  } equals
  • 2004
  • 2006
  • 2005
  • None of these
The function 
f(x) = \int_1^x [ 2 (t-1) (t-2)^3+3(t-1)^2 (t-2)^2] dt has :
  • Maximum at x=1
  • Minimum at x = \dfrac {7}{5}
  • Neither maximum nor minimum at x=2
  • All of these
Let f(x) and g(x) be two function satisfying f(x^2)+g (4-x)=4x^3, g(4-x)+g(x)=0, then the value of \int_{-4}^{4} f(x^2)dx is:
  • 512
  • 64
  • 256
  • 0
\int_{0}^{1}{\frac{dx}{x\sqrt{x}}}
  • 2
  • -2
  • 1
  • 3
The value of \displaystyle \int _{ 0 }^{ \dfrac { \pi  }{ 4 }  }{ \cos ^{ 3 }{ 2x } dx } is:
  • \cfrac { 2 }{ 3 }
  • \cfrac { 1 }{ 3 }
  • 0
  • \cfrac { 2\pi }{ 3 }
\int _{ 0 }^{ 1 }{ x{ \left[ 1-x \right]  }^{ 11 } } dx=........
  • -\cfrac { 1 }{ 132 }
  • -\cfrac { 1 }{ 156 }
  • -\cfrac { 1 }{ 121 }
  • -\cfrac { 1 }{ 12 }
What is \displaystyle \int_{0}^{2\pi}\sqrt {1 + \sin \dfrac {x}{2}}dx equal to?
  • 8
  • 4
  • 2
  • 0
The value of \displaystyle \int_{0}^{\dfrac {\pi}{4}} (\sqrt {\tan x} +  \sqrt {\cot x})\,\, dx is equal to
  • \dfrac {\pi}{2}
  • -\dfrac {\pi}{2}
  • \dfrac {\pi}{\sqrt {2}}
  • -\dfrac {\pi}{\sqrt {2}}
\int _{ a }^{ b }{ \cfrac { \log { x }  }{ x }  } dx=.......\quad (where a,b\in { R }^{ + })
  • \cfrac { 1 }{ 2 } \log { \left( ab \right) } \log { \left( \cfrac { b }{ a } \right) }
  • \cfrac { 1 }{ 2 } \log { \left( ab \right) }
  • \log { \left( \cfrac { b }{ a } \right) }
  • 2\log { \left( \cfrac { b }{ a } \right) }
\int _{ 0 }^{ \pi /2 }{ \cfrac { \cos { 2x }  }{ { \left( \sin { x } +\cos { x }  \right)  }^{ 2 } }  } dx=......
  • \cfrac { \pi }{ 4 }
  • \cfrac { \pi }{ 2 }
  • 0
  • -\cfrac { \pi }{ 4 }
Evaluate : \displaystyle\int^1_0\dfrac{dx}{\sqrt{5x+3}}
  • \dfrac{2}{5}(\sqrt{8}-\sqrt{3})
  • \dfrac{2}{5}(\sqrt{8}+\sqrt{3})
  • \dfrac{2}{5}\sqrt{8}
  • None of these
Evaluate : \displaystyle\int^2_0\sqrt{6x+4}dx
  • \dfrac{64}{9}
  • 7
  • \dfrac{56}{9}
  • \dfrac{60}{9}
Evaluate \displaystyle\int^{\pi/4}_0\tan^2xdx
  • \left(1-\dfrac{\pi}{4}\right)
  • \left(1+\dfrac{\pi}{4}\right)
  • \left(1-\dfrac{\pi}{2}\right)
  • \left(1+\dfrac{\pi}{2}\right)
Solve: \displaystyle \int _{ 0 }^{ \pi /4 }{ \tan ^{ 100 }{ x }  } dx+\int _{ 0 }^{ \pi /4 }{ \tan ^{ 102 }{ x }  } dx=....
  • \cfrac { 1 }{ 101 }
  • \cfrac { 1 }{ 102 }
  • \cfrac { 1 }{ 100 }
  • 101
If I=\displaystyle\overset{1}{\underset{0}{\displaystyle\int}}x(1-x)^{1/2}dx and 60I+k=25 then k= _________. (k\in R).
  • 9
  • 25
  • 60
  • 41
\displaystyle \int_1^{32}\dfrac{dx}{x^{1/5}\sqrt{1+x^{4/5}}}
  • \dfrac{2}{5}(\sqrt{17}+\sqrt{2})
  • \dfrac{2}{5}(\sqrt{17}-\sqrt{2})
  • \dfrac{5}{2}(\sqrt{17}-\sqrt{2})
  • \dfrac{5}{2}(\sqrt{17}+\sqrt{2})
\int _{ 0 }^{ 5 }{ \sqrt { 25-{ x }^{ 2 } }  } dx=.........
  • 25\pi
  • \cfrac { 25\pi }{ 4 }
  • \cfrac { \pi }{ 4 }
  • \cfrac { 25 }{ 4 }
\int _{ 0 }^{ \pi /2 }{ \sin { 2x } .\sin { x }  } dx=.....
  • \cfrac{1}{3}
  • \cfrac{2}{3}
  • -\cfrac{2}{3}
  • \cfrac{4}{3}
Evaluate \displaystyle \int_{-2\pi}^{5\pi} \cot^{-1} (\tan x) dx.
  • 0
  • -1
  • 1
  • 2
If \int _{ 0 }^{ \pi /3 }{ \dfrac { \cos { x }  }{ 3+4\sin { x }  } dx } =k\log { \left( \dfrac { 3+2\sqrt { 3 }  }{ 3 }  \right)  }, then, k is equal to ?
  • \dfrac{1}{2}
  • \dfrac{1}{3}
  • \dfrac{1}{4}
  • \dfrac{1}{8}
Find proper substitution
\int _{ 0 }^{ 1 }{ \dfrac { { e }^{ -x } }{ 1+{ e }^{ -x } } dx }
  • 1+{ e }^{ -x }\rightarrow t
  • -{ e }^{ -x }dx\rightarrow dt
  • -\int _{ 0 }^{ 1 }{ \dfrac { dt }{ t } }
  • -\int _{ 0 }^{ 1 }{ ln\left| t \right| }
If \displaystyle  \int _{ 2 }^{ x }{ { \left( { e }^{ x }-1 \right)  }^{ -1 }dx=\left( \dfrac { 3 }{ 2 }  \right)  } then x=
  • \ln 4
  • 1
  • {e}^{2}
  • \dfrac{1}{e}
Integrate \displaystyle \int _{ 0 }^{ 2\pi  }{ \sqrt {1+\sin x }}dx
  • 4
  • 6
  • 8
  • None of these
Evaluate the following integral \int _{ 0 }^{ \infty  }{ \dfrac { dx }{ \left( { x }^{ 2 }+{ a }^{ 2 } \right) \left( { x }^{ 2 }+{ b }^{ 2 } \right)  }  } =
  • \dfrac {\pi (a+b)}{ab}
  • \dfrac {\pi ab}{(a+b)}
  • \dfrac {\pi}{2(a+b)}
  • \dfrac {\pi}{2ab(a+b)}
If \displaystyle \int_{0}^{40} \dfrac{dx}{2x+1} =\log a, then a is:
  • 3
  • 9
  • 81
  • 40
\displaystyle \int_{2}^{4} \dfrac{\sqrt{x^{2}-4}}{x^{4}}dx=
  • \dfrac{3}{32}
  • \dfrac{\sqrt{3}}{32}
  • \dfrac{3}{8}
  • \dfrac{\sqrt{3}}{8}
Integrate \int_{0}^{1}\dfrac{\sqrt x}{1+x^3}
  • \dfrac {\pi}{4}
  • \dfrac {\pi}{3}
  • \dfrac {\pi}{6}
  • \dfrac {\pi}{2}
The value of \displaystyle \int _0^{\pi/2} \sin x \cos x dx

  • \dfrac 12
  • \dfrac 34
  • 2
  • None\ of\ these
\displaystyle \int _{ 0 }^{ 2x }{ \sqrt { 1+\sin { x }  }  } dx
  • \sin\dfrac{x}{2}-\cos\dfrac{x}{2}-\dfrac{\pi}{2}+C
  • \sin\dfrac{x}{2}-\cos\dfrac{x}{2}+\dfrac{\pi}{2}+C
  • 2\sin x - 2\cos x + 2
  • None of these
\displaystyle \int_{0}^{1}{\dfrac{ln(1+x)}{1+{x}^{2}}}dx=
  • \dfrac{\pi}{4}ln{2}
  • \dfrac{\pi}{2}ln{2}
  • \dfrac{\pi}{8}ln{2}
  • \pi\ ln{2}
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Practice Class 12 Commerce Applied Mathematics Quiz Questions and Answers