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

$$\displaystyle\int_0^\pi \frac{1}{1\, +\,\sin x}\ dx$$ 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