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

If $$f(x)=\begin{vmatrix} \cos { x }  & 1 & 0 \\ 1 & 2\cos { x }  & 1 \\ 0 & 1 & 2\cos { x }  \end{vmatrix}$$ then $$\displaystyle\int _{ 0 }^{ \pi /2 }{ f\left( x \right) } dx$$ is equal to 
  • $$1/4$$
  • $$-1/3$$
  • $$1/2$$
  • $$none of these $$
The value of $$\int_{0}^{[x]} (x-[x])dx$$, where $$[x]$$ is the greatest integer $$|le x$$ is equal to
  • $$4[x]$$
  • $$2[x]$$
  • $$\dfrac{1}{2} [x]$$
  • $$\dfrac{1}{5} [x]$$
The value of $$\int_{1}^{-1}\dfrac{dx}{(2-x)\sqrt{1-x^{2}}}$$ is 
  • $$0$$
  • $$\dfrac{\pi}{\sqrt{3}}$$
  • $$\dfrac{2\pi}{3\sqrt{3}}$$
  • cannot be evaluated
The value of $$\displaystyle \int _{0}^{\infty}\dfrac{\sqrt{x^2+1}}{(x+\sqrt{x^{2}+1})^{n+1}} .dx  \ \forall \ n\ \in \ N- \{ \pm 1 \}$$ is
  • $$0$$
  • $$n(n^{2}-1)$$
  • $$\dfrac{n}{(n^{2}-1)}$$
  • $$n^{2}$$
$$\displaystyle \int_{-3\pi}^{3\pi}{\sin^{2}\theta\sin^{2} 2\theta d \theta}$$ is equal to-
  • $$\pi$$
  • $$\dfrac{3\pi}{2}$$
  • $$\dfrac{5\pi}{2}$$
  • $$6\pi$$
Let $$f\left(x\right)+f\left(\dfrac{1}{x}\right)=F\left(x\right)$$ where $$f\left(x\right)=\displaystyle\int_{1}^{x}{\dfrac{\ln{t}}{1+t}dx}$$.Then $$F\left(e\right)=$$
  • $$\dfrac{-1}{2}$$
  • $$\dfrac{1}{2}$$
  • $$1$$
  • $$-1$$
Evaluate:
$$\displaystyle\int_{-2}^{3}{\left|1-{x}^{2}\right|dx}$$
  • $$\dfrac{28}{3}$$
  • $$\dfrac{2}{3}$$
  • $$\dfrac{8}{3}$$
  • $$\dfrac{5}{3}$$
$$\displaystyle \int_{0}^{\pi/2}{(\sin x-\cos x).\log(\sin x+\cos x)dx}$$=
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{2}$$
  • $$0$$
  • $$2$$
$$\displaystyle \overset{2\pi}{\underset{0}{\int}} x \,log \left(\dfrac{3 + \cos x}{3 - \cos x}\right)dx$$
  • $$\dfrac{\pi}{2} \,log \,3$$
  • $$\dfrac{\pi}{12} \,log \,3$$
  • $$\dfrac{\pi}{3} \,log \,3$$
  • $$0$$
$$\int^2_0 (\sqrt{\dfrac{4-x}{x}} - \sqrt{\dfrac{x}{4-x}})dx$$ is equal to
  • $$0$$
  • $$8$$
  • $$4$$
  • $$16$$
If $$2\displaystyle \int_{0}^{1}\tan^{-1}xdx=\displaystyle \int_{0}^{1}\cot^{-1}(1-x+x^{2})dx$$, then $$\displaystyle \int_{0}^{1}\tan^{-1}(1-x+x^{2})dx$$ is equal to: 
  • $$\log 4$$
  • $$\dfrac {\pi}{2}+\log 2$$
  • $$\log 2$$
  • $$\dfrac {\pi}{2}-\log 4$$
The value of $${\int }_{0}^{1}\dfrac{dx}{x+\sqrt{1-x^{2}}}$$ is
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{\pi}{4}$$
Let $$f: { (2,3)\rightarrow (0,1) }$$ be defined by $$f{ (x)=x-[x] }$$ then $$f^{ -1 }{ (x) }$$ equals
  • $$2x+1$$
  • $$x+1$$
  • $$x-1$$
  • $$x+2$$
If $$I=\displaystyle\int _{ 8 }^{ 15 }{ \dfrac { dx }{ \left( x-3 \right) \sqrt { x+1 }  }  } $$, then $$I$$ equals 
  • $$\dfrac{1}{2}\log\dfrac{5}{3}$$
  • $$2\log\dfrac{1}{3}$$
  • $$\dfrac{1}{2}\log\dfrac{1}{5}$$
  • $$2\log\dfrac{5}{3}$$
$$\displaystyle \int_{2}^{8}\dfrac {\sqrt {10-x}}{\sqrt {x}+\sqrt {10-x}}dx$$ is
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
$${\int}_{0}^{\pi}\dfrac{\sqrt{1-x}}{1+x}dx=$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{2}-1$$
  • $$\dfrac{\pi}{2}+1$$
  • $$\pi+1$$
$$\int_{1}^{\infty }(e^{x+1}+e^{3-x})^{-1}dx$$ is equal to: 
  • $$\frac{\pi }{4e^{2}}$$
  • $$\frac{\pi }{4e}$$
  • $$\frac{1}{e^{2}}(\frac{\pi }{2}-tan^{-1}\frac{1}{e})$$
  • $$\frac{\pi }{2e^{2}}$$
$$\int_{0}^{\frac{1}{2}}\frac{xsin^{-1}x}{\sqrt{1-x^{2}}}dx$$ is equal to
  • $$\frac{1}{2}+\frac{\pi }{2\sqrt{3}}$$
  • $$\frac{1}{2}-\frac{\pi }{2\sqrt{3}}$$
  • $$\frac{1}{2}+\frac{\pi }{4\sqrt{3}}$$
  • $$\frac{1}{2}-\frac{\pi }{4\sqrt{3}}$$
The integral $$\displaystyle \int_{\dfrac{\pi}{12}}^{\dfrac{\pi}{4}}{\dfrac{8\cos 2x}{(\tan x+\cot x)^{3}}dx}$$ equals :
  • $$\dfrac{15}{128}$$
  • $$\dfrac{15}{64}$$
  • $$\dfrac{13}{32}$$
  • $$\dfrac{13}{256}$$
If $$P=\displaystyle \lim _{ n\rightarrow \infty  }{ \frac { { \left( \prod _{ r=1 }^{ n }{ \left( { n }^{ 3 }+{ r }^{ 3 } \right)  }  \right)  }^{ 1/n } }{ { n }^{ 3 } }  }$$  and $$\lambda =\displaystyle \int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 3 } }  } $$ then $$In P$$ is equal to
  • $$In 2-1+\lambda$$
  • $$In 2-3+3\lambda$$
  • $$2In 2-\lambda$$
  • $$In 4-3+3\lambda$$
$$\displaystyle \overset{3}{\underset{0}{\int}} \dfrac{dx}{\sqrt{5 - x^2}}$$
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{2}$$
  • $$-\dfrac{\pi}{2}$$
  • $$-\dfrac{\pi}{6}$$
$$\displaystyle\int _{ 0 }^{ \infty  }{ \dfrac { dx }{ \left( x+\sqrt { { x }^{ 2 }+1 }  \right) ^{ 3 } }  } =$$
  • $$\dfrac{3}{8}$$
  • $$\dfrac{1}{8}$$
  • $$-\dfrac{3}{8}$$
  • $$None\ of\ these$$
If $${ I }_{ 1 }=\int _{ 0 }^{ \pi /2 }{ \cos(\sin x)dx, { I }_{ 2 }= } \int _{ 0 }^{ \pi /2 }{ \sin(\cos x)dx } $$ and $${ I }_{ 3 }=\int _{ 0 }^{ \pi /2 }{ \cos xdx } $$, then
  • $${ I }_{ 1 }>{ I }_{ 2 }>{ I }_{ 3 }$$
  • $${ I }_{ 3 }>{ I }_{ 2 }>{ I }_{ 1 }$$
  • $${ I }_{ 3 }>{ I }_{ 1 }>{ I }_{ 2 }$$
  • $${ I }_{ 1 }>{ I }_{ 3 }>{ I }_{ 2 }$$
If I =$$\overset { 2 }{ \underset { -3 }{ \int }  } (|x + 1| + |x + 2| +|x -1|) dx$$, then i equals 
  • $$\dfrac{31}{2}$$
  • $$\dfrac{35}{2}$$
  • $$\dfrac{47}{2}$$
  • $$\dfrac{39}{2}$$
If $$f(x)\int _{ 1 }^{ x }{ \frac { { tan }^{ 1 }t }{ t }  } dt(x>0)$$, then the value of $$f({ o }^{ 2 })-f(\frac { 1 }{ o^{ 2 } } )$$
  • $$\pi $$
  • $$2\pi $$
  • $$\frac { \pi }{ 2 } $$
  • 0
The floor value of integral $$\displaystyle \int_\dfrac{\pi }{4}^{3\pi }\dfrac{x}{1+4x}dx$$ is 
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
Let $$I_{1}=\int_{-2}^{2} \dfrac{x^{6}+3 x^{5}+7 x^{4}}{x^{4}+2} d x$$ and$$I_{2}=\int_{-3}^{1} \dfrac{2(x+1)^{2}+11(x+1)+14}{(x+1)^{4}+2} d x,$$ then the value of$$I_{1}+I_{2}$$ is
  • 8
  • $$200 / 3$$
  • $$100 / 3$$
  • None of these
If $$I_1=\displaystyle\int^1_x\dfrac{dt}{1+t^2}$$ and $$I_2=\displaystyle\int^{1/x}_1\dfrac{dt}{1+t^2}$$ for $$x > 0$$, then?
  • $$I_1 = I_2$$
  • $$I_1 > I_2$$
  • $$I_2 > I_1$$
  • $$I_2=\cot^{-1}x-\pi/4$$
If $$z=x+3i$$ then value of $$\displaystyle\int^4_2\left[arg\left|\dfrac{z-i}{z+i}\right|\right]dx$$, where $$[.]$$ denotes the greatest integer function, is?
  • $$3\sqrt{2}$$
  • $$6\sqrt{3}$$
  • $$\sqrt{6}$$
  • None
Suppose $$I_1=\displaystyle \int_{0}^{\pi/2} \cos(\pi \sin^2 x)dx;I_2=\displaystyle \int_{0}^{\pi/2} \cos(2\pi \sin^2x)dx$$ and $$I_3=\displaystyle \int_{0}^{\pi/2} \cos(\pi \sin x)dx $$ then
  • $$I_1=0$$
  • $$I_2+I_3=0$$
  • $$I_1+I_2+I_3=0$$
  • $$I_2=I_3$$
If m,n $$\in $$N, then the value of $$\displaystyle \int_{a}^{b}(x-a)^m (b-x)^n dx$$is equal to 
  • $$\dfrac{(b-a)^{m+n}.m!n!}{(m+n)!}$$
  • $$\dfrac{(b-a)^{m+n+1}.m!n!}{(m+n+1)!}$$
  • $$\dfrac{(b-a)^{m}.m!}{m!}$$
  • None of these
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