CBSE Questions for Class 12 Commerce Maths Integrals Quiz 2 - MCQExams.com

The integral $$\displaystyle \int_{0}^{1}\frac{(\mathrm{t}\mathrm{a}\mathrm{n}^{-1}{\mathrm{x})^{3}}}{1+\mathrm{x}^{2}}\mathrm{d}\mathrm{x}=$$
  • $$\displaystyle \frac{\pi^{4}}{64}$$
  • $$\displaystyle \frac{\pi^{4}}{256}$$
  • $$\displaystyle \frac{\pi^{4}}{1024}$$
  • $$\displaystyle \frac{\pi^{4}}{512}$$

The integral $$\displaystyle \int_{0}^{\pi/4} \displaystyle \frac{e^{\tan x}}{\cos^{2}x}dx=$$
  • $$e-1$$
  • $${e}^{-1}-1$$
  • $${e}^{-1}+1$$
  • $${e}^{-2}-1$$

$$\displaystyle \int_{0}^{1/2}\mathrm{e}^{\mathrm{x}}\left[ { s }{ i }{ n }^{ -1 }{ x }+\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } }  }  \right] $$ dx $$=$$


  • $$\displaystyle \frac{e^{4}}{4}$$
  • $$\displaystyle \frac{\pi\sqrt{\mathrm{e}}}{6}$$
  • $$\displaystyle \frac{\sqrt{\pi \mathrm{e}}}{4}$$
  • $$\displaystyle \frac{\pi\sqrt{\mathrm{e}}}{2}$$
If  $$\displaystyle \int_{0}^{k}\frac{\cos x}{1+\sin^{2}x}dx=\frac{\pi}{4}$$ then $${k}=?$$
  • $$1$$
  • $$\pi/4$$
  • $$\pi/2$$
  • $$\pi/6$$

$$\displaystyle \int_{0}^{1}\tanh xdx=$$
  • $$\log(\mathrm{e}+1/\mathrm{e})$$
  • $$\log$$ $$(e-1/e)$$
  • $$\log(\mathrm{e}/2+1/2\mathrm{e})$$
  • $$\displaystyle \log(\frac{\mathrm{e}}{2}-\frac{1}{\mathrm{e}})$$
Evaluate the integral
$$\displaystyle \int_{1}^{e}\frac{(\ln x)^{3}}{x}dx $$
  • $$e^{4}/4$$
  • $$\dfrac{1}{4}$$
  • $$\displaystyle \frac{1}{4}({e}^{4}-1)$$
  • $$e^{4}-1$$

$$\displaystyle \int_{0}^{\pi}\frac{\tan x}{\sec x+\cos x}dx_{=}$$
  • $$\pi$$
  • $$\dfrac {-\pi}2$$
  • $$-\pi$$
  • $$2 \pi$$
Evaluate: $$\displaystyle \int_{\sqrt{8}}^{\sqrt{15}}x\sqrt{1+x^{2}}.dx$$
  • $$\dfrac{15}{8}$$
  • $$\dfrac{37}{3}$$
  • $$\dfrac{37}{6}$$
  • $$\displaystyle \frac{37}{9}$$

$$\displaystyle \int_{0}^{1}\mathrm{e}^{\mathrm{x}}(\mathrm{e}^{\mathrm{x}}+1)^{3}\mathrm{d}\mathrm{x}=$$
  • $$\displaystyle \frac{e^{4}}{4}-4$$
  • $$\displaystyle \frac{(e+1)^{4}}{4}-4$$
  • $$\displaystyle \frac{(e+1)^{4}+16}{4}$$
  • $$\displaystyle \frac{(\mathrm{e}+1)^{4}}{4}+4$$

$$\displaystyle \int_{0}^{1}\frac{xdx}{(x^{2}+1)^{2}}=$$
  • $$1/2$$
  • $$1/3$$
  • $$1/4$$
  • $$0$$
Evaluate: $$\displaystyle \int_{0}^{\dfrac{\pi}{2}}e^{\sin^2 x}\sin 2xdx$$
  • $$e$$
  • $$e+1$$
  • $$e-1$$
  • $$2{e}+1$$

$$\displaystyle \int_{0}^{4}\sqrt{16-x^{2}}d_{X}=$$
  • $$\displaystyle \frac{\pi}{4}$$
  • $$\pi$$
  • $$ 16\pi$$
  • $$ 4\pi$$

Find $$\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sec^{2}xdx}{(\sec x+\tan x)^{n}}$$, where $$(\mathrm{n}>2)$$
  • $$\displaystyle \frac{1}{n^{2}-1}$$
  • $$\displaystyle \frac{n}{n^{2}-1}$$
  • $$\displaystyle \frac{n}{n^{2}+1}$$
  • $$\displaystyle \frac{2}{n^{2}-1}$$
$$\displaystyle \int_{0}^{\pi/2}\frac{d_{X}}{4\cos^{2}x+9\sin^{2}x}=$$
  • $$\displaystyle \frac{\pi}{12}$$
  • $$\displaystyle \frac{\pi}{4}$$
  • $$\displaystyle \frac{\pi}{9}$$
  • $$\displaystyle \frac{\pi}{6}$$
Evaluate the following definite integral:
$$\displaystyle \int_{0}^{\pi_/{2}}\frac{1}{4+5\cos x}dx=$$
  • $$\displaystyle \frac{1}{5}\log 2$$
  • $$\displaystyle \frac{1}{2}\log 2$$
  • $$\displaystyle \frac{1}{3}\log 3$$
  • $$\displaystyle \frac{1}{3}\log 2$$
$$\displaystyle \int_{0}^{1}\frac{1}{\sqrt{2+3x}}dx_{=}$$
  • $$\displaystyle \frac{2}{3}(\sqrt{5}-\sqrt{2})$$
  • $$\displaystyle \frac{2}{3}(\sqrt{5}+\sqrt{2})$$
  • $$\displaystyle \frac{3}{5}(\sqrt{5}-\sqrt{2})$$
  • $$\displaystyle \frac{2}{3}(\sqrt{3}-\sqrt{2})$$

$$\displaystyle \int_{0}^{\infty}(a^{-x}-b^{-x})dx=(\mathrm{a}>1,\mathrm{b}>1)$$
  • $$\displaystyle \dfrac{1}{\log a}-\dfrac{1}{\log b}$$
  • $$\log a-\log b$$
  • $$\log a + \log b$$
  • $$\displaystyle \dfrac{1}{\log a}+\dfrac{1}{\log b}$$

$$\displaystyle \int_{0}^{a}\frac{x-a}{x+a} dx =$$
  • $$a+2a\log 2$$
  • $$a - 2a\log 2$$
  • $$2a\log-a$$
  • $$2a\log 2$$

If $$\displaystyle \int_{0}^{60}\frac{dx}{2x+1}=\log   a$$, then $$ a= $$
  • $$3$$
  • $$11$$
  • $$81$$
  • $$40$$
Evaluate the integral
$$\displaystyle \int_{0}^{1}\frac{1-x}{1+x}dx$$
  • $$\log 4$$
  • $$\log (\dfrac{4}{e})$$
  • $$1$$
  • $$\log (\dfrac{e}{4})$$

$$\displaystyle \int_{1}^{2}\frac{\mathrm{d}\mathrm{x}}{\sqrt{1+\mathrm{x}^{2}}}=$$
  • $$\displaystyle \log_{\mathrm{e}}(\frac{2+\sqrt{5}}{\sqrt{2}+1})$$
  • $$\displaystyle \log_{\mathrm{e}}(\frac{\sqrt{2}+1}{2+\sqrt{5}})$$
  • $$\displaystyle \log_{\mathrm{e}}(\frac{2-\sqrt{5}}{\sqrt{2}-1})$$
  • 0
$$\int_{-1}^{1} \displaystyle \frac{d{x}}{1+x^{2}}=$$
  • 0
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\displaystyle \frac{\pi}{4}$$
  • $$\displaystyle \frac{\pi}{6}$$
Evaluate the integral
$$\displaystyle \int_{1}^{2}\sqrt{(x-1)(2-x)}dx $$
  • $$\dfrac {\pi }{8}$$
  • $$\dfrac {\pi }{4}$$
  • $$\dfrac {1}{8}$$
  • $$\dfrac {1}{4}$$

$$\displaystyle \int_{1}^{\infty}\left( \frac { 1 }{ 1+x^{ 2 } }  \right) d{ x }=$$

  • $$\displaystyle \frac{\pi}{4}$$
  • $$-\displaystyle \frac{\pi}{4}$$
  • $$\displaystyle \frac{\pi}{2}$$
  • $$-\displaystyle \frac{\pi}{2}$$
Evaluate: $$\displaystyle \int_{0}^{\pi /8} \cos^{3}4x\ dx $$
  • $$1/6$$
  • $$1/5$$
  • $$-1/3$$
  • $$1/8$$
$$\displaystyle \int \sec^{2}x.\text{cosec}^{2}xdx=$$
  • $$ \tan x-\cot x +c$$
  • $$\tan x +\cot x +c$$
  • $$-\tan x + \cot x +c$$
  • $$\sec x \tan x +c$$
$$\int_{0}^{\pi /2} \sin^{4}x.\cos^{2}xd_{X=}$$
  • $$\displaystyle \frac{\pi}{32}$$
  • $$\displaystyle \frac{\pi^{2}}{16}$$
  • $$\displaystyle \frac{\pi}{15}$$
  • $$\displaystyle \frac{\pi}{64}$$
Evaluate the integral
$$\displaystyle \int_{1}^{3}\frac{d_{X}}{\sqrt{(x-1)(3-x)}}$$
  • $$\pi$$
  • $$-\pi$$
  • $$\pi/2$$
  • $$0$$

$$\displaystyle \int_{0}^{\infty}\frac{dx}{(x+\sqrt{x^{2}+1})^{5}}=$$
  • 1/24
  • 1/5
  • 5/24
  • 5/36
Evaluate : $$\displaystyle \int\frac{\cot^{2}x}{(co\sec^{2}x+co\sec x)}d{x}$$
  • $$ x-\sin x +C$$
  • $$x + \cos x +C$$
  • $$ \sin x - x+C  $$
  • $$2\tan(\displaystyle \frac{ax}{2})+c$$
$$\displaystyle \int_{0}^{1}\sqrt{x(1-x)}dx=$$
  • $$\pi /2$$
  • $$\pi /4$$
  • $$\pi /6$$
  • $$\pi /8$$
Evaluate the integral
$$\displaystyle\int_{\pi}^{5\pi/4}\frac{\sin 2x}{\cos^4 x + \sin^4 x}dx $$
  • $$\displaystyle \frac{5\pi}{4}$$
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\pi$$
  • $$\displaystyle \frac{\pi}{4}$$

$$\displaystyle \int_{0}^{a}\frac{x^{5}dx}{\sqrt{a^{2}-x^{2}}}=$$
  • $$\displaystyle \frac{\mathrm{a}^{5}}{15}$$
  • $$\displaystyle \frac{8\mathrm{a}^{5}}{15}$$
  • $$\displaystyle \frac{8a}{15}$$
  • $$\displaystyle \frac{11\mathrm{a}^{2}}{15}$$
Evaluate the integral
$$\displaystyle \int_{0}^{1}  cos ^{-1}\left(\displaystyle \frac{1- {x}^{2}}{1+ {x}^{2}}\right) {d} {x}$$
  • $$\displaystyle \frac{\pi}{2} -\log 2$$
  • $$\displaystyle \frac{\pi}{2}+\log 2$$
  • $$\displaystyle \frac{\pi}{4}$$ - log 2
  • $$\displaystyle \frac{\pi}{4}$$ - log 3
Evaluate: $$\displaystyle \int_{1/3}^{1}\frac{(x-x^{3})^{1/3}}{x^{4}}dx$$.
  • $$3$$
  • $$0$$
  • $$6$$
  • $$4$$

$$\displaystyle \int_{\log 2}^{t}\frac{d_{X}}{\sqrt{e^{x}-1}}=\frac{\pi}{6}$$, then $$\mathrm{t}=$$
  • 4
  • $$\log 8$$
  • log 4
  • log 2
Evaluate: $$\displaystyle \int_{0}^{\pi /2} \sin^{3}x.\cos^{3}x dx$$
  • $$\displaystyle \frac{1}{12}$$
  • $$\displaystyle \frac{\pi}{24}$$
  • $$\displaystyle \frac{\pi}{12}$$
  • $$\displaystyle \frac{1}{24}$$
Evaluate: $$\displaystyle \int_{0}^{1} \cos$$ $$\left(2 \cot^{-1}\sqrt{\displaystyle \frac{1- {x}}{1+ {x}}}\right)dx $$
  • $$\dfrac{-1}{2}$$
  • $$\dfrac{1}{2}$$
  • $$0$$
  • $$1$$
Evaluate the integral
$$I=\displaystyle \int _{ 0 }^{ \frac { 1 }{ \sqrt { 2 }  }  }{ \cfrac { { sin }^{ -1 }x }{ { \left( 1-{ x }^{ 2 } \right)  }^{ \frac { 3 }{ 2 }  } } dx }  $$
  • $$\displaystyle \frac{\pi}{4}+\frac{1}{2}$$ log 2
  • $$\displaystyle \frac{\pi}{4}-\frac{1}{2}$$ log 2
  • $$\displaystyle \frac{\pi}{3}$$
  • $$\displaystyle \frac{\pi}{6}$$

lf $$0<\mathrm{a}<\mathrm{c},\ 0<\mathrm{b}<\mathrm{c}$$ then $$\displaystyle \int_{0}^{\infty}\frac{a^{x}-b^{x}}{c^{x}}dx=$$
  • $$\displaystyle \log\frac{b}{c}-\log\frac{a}{c}$$
  • $$\displaystyle \frac{\log a-\log b}{\log c}$$
  • $$\displaystyle \frac{1}{\log b/c}-\frac{1}{\log a/c}$$
  • $$l\displaystyle \mathrm{o}\mathrm{g}\frac{\mathrm{a}}{\mathrm{c}}-l\mathrm{o}\mathrm{g}\frac{\mathrm{b}}{\mathrm{c}}$$

$$\displaystyle \int_{0}^{1}\frac{xe^{x}}{(x+1)^{2}}dx=$$
  • $$\displaystyle \frac{e}{2}$$
  • $$\displaystyle \frac{e-1}{2}$$
  • $$\displaystyle \frac{e}{2}-1$$
  • $$\displaystyle \frac{e-3}{2}$$

The solution of the equation $$\displaystyle \int_{\sqrt{2}}^{\mathrm{x}}\frac{\mathrm{d}\mathrm{x}}{\mathrm{x}\sqrt{\mathrm{x}^{2}-1}}=\frac{\pi}{12}$$ is
  • 1
  • 1/2
  • 2
  • -2
Evaluate the integral
$$\displaystyle \int_{0}^{1}\frac{ {d} {x}}{ {x}^{2}+2 {x} {c} {o} {s}\alpha+1}$$
  • $$\sin \alpha$$
  • $$\tan^{-1}$$ $$(\sin \alpha)$$
  • $$\dfrac{\alpha}{(2\sin \alpha)}$$
  • $$\alpha$$ $$(\sin \alpha)$$

$$\displaystyle \int_{0}^{\pi/2}\frac{1}{1+4\sin^{2}x}dx=$$
  • $$\displaystyle \frac{\pi}{\sqrt{5}}$$
  • $$\displaystyle \frac{\pi}{2\sqrt{5}}$$
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\displaystyle \frac{\pi}{3\sqrt{5}}$$

$$\displaystyle \int_{0}^{16}\frac{dx}{\sqrt{x+9}-\sqrt{x}}=$$
  • 10
  • 12
  • 14
  • 16

$$\displaystyle \int_{0}^{1}\frac{\sqrt{x}}{1+x}dx_{=}$$
  • $$2-\pi/2$$
  • $$1-\pi/2$$
  • $$\pi/2$$
  • $$2+\pi/2$$

$$\displaystyle \int_{0}^{\pi/4}\frac{\sqrt{\tan x}}{sin x cos x}dx=$$
  • 1
  • 2
  • 0
  • 4

$$\displaystyle \int_{1}^{2}\frac{dx}{x^{2}-2x+4}=$$
  • 0
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\displaystyle \frac{\pi}{3}$$
  • $$\displaystyle \frac{\pi}{6\sqrt{3}}$$

$$\displaystyle \int_{0}^{1}\frac{x}{1+\sqrt{x}}dx_{=}$$
  • $$\frac{5}{3}-\log 4$$
  • $$\frac{5}{3}+\log 4$$
  • $$\frac{5}{3}\log 4$$
  • $$\displaystyle \frac{3}{5}-l\mathrm{o}\mathrm{g}4\frac{3}{5}-l\mathrm{o}\mathrm{g}4$$

$$\displaystyle \int_{0}^{\pi}\frac{dx}{3+2\sin x+\cos x}=$$
  • $$\pi/3$$
  • $$\pi/4$$
  • $$\pi/6$$
  • $$\pi/2$$
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