CBSE Questions for Class 12 Commerce Applied Mathematics Definite Integrals Quiz 6 - MCQExams.com

$$\int _{ 0 }^{ \pi  }{ x\ln { \left( \sin { x }  \right)  }  } dx=$$
  • $$\dfrac { \pi }{ 2 } \ln { 2 } $$
  • $$\dfrac { -\pi^{2} }{ 2 } \ln { 2 } $$
  • $$\dfrac { -\pi }{ 2 } \ln { 2 } $$
  • $$-2p\ \ln { 2 } $$
Evaluate $$\displaystyle\int^{\pi}_0\dfrac{dx}{(1+\sin x)}$$
  • $$\dfrac{1}{2}$$
  • $$1$$
  • $$2$$
  • $$0$$
Evaluate $$\displaystyle\int^2_0\dfrac{dx}{\sqrt{4-x^2}}$$
  • $$1$$
  • $$\sin^{-1}\dfrac{1}{2}$$
  • $$\dfrac{\pi}{4}$$
  • None of these
$$\displaystyle \int_{0}^{1}\sin \left(2\tan^{-1}\sqrt{\dfrac{1+x}{1-x}}\right)dx=$$
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{2}$$
  • $$\pi$$
$$\int_{0}^{2\pi}{ln(1+\cos{x})}dx=$$
  • $$\pi\ln{2}$$
  • $$-\pi\ln{2}$$
  • $$-2\pi\ln{2}$$
  • $$2\pi\ln{2}$$
Evaluate $$\displaystyle \int_\cfrac{\pi}{4}^\cfrac{\pi}{2}(\sqrt{\tan x}+\sqrt{\cot x})dx=$$
  • $$\dfrac{\pi}{2\sqrt{2}}$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{\sqrt{2}}$$
  • $$\dfrac{\pi}{3}$$
$$\displaystyle \int _0^4\dfrac{2x+3}{x^2+3x+2}dx$$
  • $$\log 3$$
  • $$\log 5$$
  • $$\log 15$$
  • $$\log 2$$
Let $$I = \int\limits_1^3 {\sqrt {{x^4} + {x^2}} dx,} $$then I= 
  • $$1 > 6\sqrt {10} $$
  • $$1 < 2\sqrt 2 $$
  • $$2\sqrt 2 < 1 < 6\sqrt {10} $$
  • none of above
$$\int_\limits {0}^\cfrac{\pi}{4}\dfrac{1}{1+2\sin^2x}dx$$
  • $$\dfrac{\pi}{3\sqrt3}$$
  • $$-\dfrac{\pi}{3\sqrt3}$$
  • $$-\dfrac{\pi}{2\sqrt3}$$
  • $$-\dfrac{\pi}{\sqrt3}$$
$$\displaystyle\int^{\pi/4}_0\sec^4xdx$$.
  • $$\dfrac{3}{4}$$
  • $$\dfrac{-3}{4}$$
  • $$\dfrac{4}{3}$$
  • $$\dfrac{5}{4}$$
For $$x>0$$, let $$f(x)=\displaystyle\int_{1}^{x}\dfrac {\log t}{1+t}dt$$ . Then $$f(x)+f\left(\dfrac {1}{x}\right)$$ is equal to:
  • $$\dfrac {1}{4}(\log x)^{2}$$
  • $$\dfrac {1}{2}(\log x)^{2}$$
  • $$\log x$$
  • $$\dfrac {1}{4}\log x^{2}$$
Evaluate $$\displaystyle\int^1_0\dfrac{x^3}{(1+x^8)}dx$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{8}$$
  • $$\dfrac{\pi}{16}$$
The value of $$\int _{ 0 }^{ \dfrac { \pi  }{ 2 }  }{ \log { \left( \dfrac { 4+3\sin { x }  }{ 4+3\cos { x }  }  \right) dx }  }$$ is
  • $$2$$
  • $$\dfrac{3}{4}$$
  • $$0$$
  • $$-2$$
$$\displaystyle \int _{ 0 }^{ \pi /4 }{ \tan ^{ 2 }{ x } dx= } $$
  • $$1-\dfrac {\pi}{4}$$
  • $$1+\dfrac {\pi}{4}$$
  • $$\dfrac {-\pi}{4}-1$$
  • $$\dfrac {\pi}{4}-1$$
The value of $$\displaystyle \underset{0}{\overset{x}{\int}} \dfrac{(t - |t|)^2}{(1 + t^2)} dt$$ is equal to 
  • $$4(x - \tan^{-1} x), \, \ \text{if} \, x < 0$$
  • $$0 \, if \, x > 0$$
  • $$\ln ( 1 + x^3) \, \ \text{if} \, x > 0 $$
  • $$4(x + \tan^{-1} x ) \, \text{if} \, x < 0$$
The integral $$\int _{ { \pi  }/{ 12 } }^{ { \pi  }/{ 4 } }{ \cfrac {

8\cos { 2x }  }{ { \left( \tan { x } +\cot { x }  \right)  }^{ 3 } } dx }

$$  equals:
  • $$\cfrac { 15 }{ 128 } $$
  • $$\cfrac { 15 }{ 64 } $$
  • $$\cfrac { 13 }{ 32 } $$
  • $$\cfrac { 13 }{ 256 } $$
If $$\int _{ 0 }^{ 1 }{ \cfrac { \tan ^{ -1 }{ x }  }{ x }  } dx=k\int _{ 0 }^{ \pi /2 }{ \cfrac { x }{ \sin { x }  }  } dx$$ then the value of $$k$$ is
  • $$1$$
  • $$\cfrac{1}{4}$$
  • $$4$$
  • $$\cfrac{1}{2}$$
The solution for x of the equation $$\displaystyle\int^x_{\sqrt{2}}\dfrac{dt}{t\sqrt{t^2-1}}=\dfrac{\pi}{2}$$ is?
  • $$- \sqrt2$$
  • $$\pi$$
  • $$\sqrt{3}/2$$
  • $$2\sqrt{2}$$
$$\displaystyle\int^{\pi}_0\sqrt{2}(1+\cos x)^{7/2}dx=?$$
  • $$\dfrac{32}{35}$$
  • $$\dfrac{64}{35}$$
  • $$\dfrac{256}{35}$$
  • $$\dfrac{512}{35}$$
Solve $$\int\limits_l^e {\dfrac{{dx}}{{\ln \left( {{x^x} \cdot {e^x}} \right)}}} $$ 
  • $$\ln 2$$
  • $$2\ln 2$$
  • $$-\ln 2$$
  • None of these
The value of $$\int _{ 0 }^{ \infty  }{ \cfrac { \log { x }  }{ 1+{ x }^{ 2 } }  } dx$$, equals
  • $$\cfrac{\pi}{2}$$ $$\log{2}$$
  • $$-\cfrac{\pi}{2}$$ $$\log{2}$$
  • $$0$$
  • $$\cfrac{\pi}{4}$$ $$\log{2}$$
Evaluate:
$$\displaystyle\int_{0}^{\pi/2}\dfrac {\sin x-\cos x}{1+\sin x\cos x}dx$$ 
  • $$0$$
  • $$1$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{4}$$
Evaluate $$\displaystyle\int^1_0\dfrac{xe^x}{(1+x)^2}dx$$
  • $$\left(\dfrac{e}{2}-1\right)$$
  • $$(e-1)$$
  • $$e(e-1)$$
  • None of these
Evaluate $$\displaystyle\int^{\pi/2}_{\pi/3} cosec xdx$$
  • $$\dfrac{1}{2} log 2$$
  • $$\dfrac{1}{2} log 3$$
  • $$-log 2$$
  • None of these
$$\displaystyle \overset{\ln \pi}{\underset{\ln\pi - \ln2}{\int}} \dfrac{e^x}{1 - \cos \left(\tfrac{2}{3}e^x\right)}dx$$ is equal to
  • $$\sqrt{3}$$
  • $$-\sqrt{3}$$
  • $$\dfrac{1}{\sqrt{3}}$$
  • $$-\dfrac{1}{\sqrt{3}}$$
Solve $$\displaystyle\int\limits_0^{\dfrac{{\ln 3}}{2}} {\dfrac{{{e^x} + 1}}{{{e^{2x}} + 1}}dx} $$
  • $$\dfrac{\pi }{{2}} + \dfrac{1}{2}ln\left( {\dfrac{3}{2}} \right)$$
  • $$\dfrac{\pi }{{12}} + \dfrac{1}{2}ln\left( {\dfrac{3}{2}} \right)$$
  • $$\dfrac{\pi }{{12}}-+ \dfrac{1}{2}ln\left( {\dfrac{3}{2}} \right)$$
  • None of these 
$$\int (1 + 2x + 3x^2 + 4x^3 + ...)dx =$$
  • $$(1 + x)^{-1} + c$$
  • $$(1 - x)^{-1} + c$$
  • $$(1 - x)^{-1} - 1 + c$$
  • None of these
Evaluate $$\displaystyle\int^{\pi}_0\dfrac{x}{1+\sin x}dx$$.
  • $$x\tan x - ln|\cos x| - x\sec x -ln|\sec x - \tan x| + C$$
  • $$x\tan x + ln|\cos x| - x\sec x -ln|\sec x - \tan x| + C$$
  • $$\tan x + ln|\cos x| - x\sec x -ln|\sec x - \tan x| + C$$
  • None of these
Evaluate $$\displaystyle\int^{\pi/3}_{\pi/6}\dfrac{dx}{1+\sqrt{\tan x}}$$.
  • $$\cfrac{\pi}{12}$$
  • $$\cfrac{7\pi}{12}$$
  • $$\cfrac{5\pi}{12}$$
  • None of these
Obtain $$ \displaystyle \int _{ 0 }^{ \pi  }{ \sqrt { 1+\cos { 2x }  } dx } $$
  • $$2$$
  • $$12$$
  • $$0$$
  • None of these
$$\displaystyle \underset{0}{\overset{\pi}{\int}} \dfrac{x \, dx}{1 + \sin \, x} = $$
  • $$\dfrac{\pi}{6}$$
  • $$\pi$$
  • $$\dfrac{\pi}{3}$$
  • none of these
$$\displaystyle\int^1_0\dfrac{2x}{\sqrt{1-x^4}}dx$$ is equal to?
  • $$\pi$$
  • $$\dfrac{\pi}{2}$$
  • $$2\pi$$
  • $$0$$
Evaluate $$\displaystyle\int^{\pi/2}_{\pi/4}\cot xdx$$
  • $$log 2$$
  • $$2 log 2$$
  • $$\dfrac{1}{2}log 2$$
  • None of these
The value of the integral $$\displaystyle\int^{1}_{\dfrac{1}{3}}\dfrac{(x-x^3)^{\dfrac{1}{3}}}{x^4}dx$$ is?
  • $$6$$
  • $$0$$
  • $$3$$
  • $$4$$
$$\int {\frac{{{{\sin }^{ - 1}}x}}{{{{\left( {1 - {x^2}} \right)}^{\frac{3}{2}}}}}dx} $$
  • $$\frac{{x\left( {{{\sin }^{ - 1}}x} \right)}}{{\sqrt {1 - {x^2}} }} + \frac{1}{2}\log \left| {\left( {1 - {x^2}} \right)} \right| + C.$$
  • $$\frac{1}{2}\log \left| {\left( {1 - {x^2}} \right)} \right| + C$$
  • $$\frac{{x\left( {{{\sin }^{ - 1}}x} \right)}}{{\sqrt {1 - {x^2}} }}+C$$
  • $$4+\frac { \pi }{ 2 } $$
$$\displaystyle\int \limits_{ -1 }^{ 1 }|x| dx = a$$ then -
  • $$a=1$$
  • $$a=2$$
  • $$a=3$$
  • $$a=4$$
What is the value of $$\int_{0}^{a}\dfrac{x-a}{x+a}\ dx$$?
  • $$a+2a\log 2$$
  • $$a-2a\log 2$$
  • $$2a\log 2-a$$
  • $$2a\log 2$$
The value of $$\int\limits_0^{\pi /2} {\dfrac{{x\sin x\cos x}}{{{{\sin }^4}x + {{\cos }^4}x}}dx\,is} $$ is 
  • $$\dfrac{{{\pi ^2}}}{4}$$
  • $$\dfrac{{{\pi ^2}}}{8}$$
  • $$\dfrac{{{\pi ^2}}}{{16}}$$
  • $$\dfrac{{3{\pi ^2}}}{{16}}$$
$$\displaystyle \int_{1/e}^{e}{|\ln x|dx}$$ equals
  • $$e^{-1}-1$$
  • $$2\left (1-\dfrac {1}{e}\right)$$
  • $$1-\dfrac {1}{e}$$
  • $$e-1$$
$$\int _{ 0 }^{ \pi /4 }{ x.\sec ^{ 2 }{ x } dx=? }$$
  • $$\frac {\pi}{4}+\log {\sqrt {2}}$$
  • $$\frac {\pi}{4}-\log {\sqrt {2}}$$
  • $$1+\log {\sqrt {2}}$$
  • $$1-\frac{1}{2}\log {2}$$
$$\displaystyle \underset{0}{\overset{\pi}{\int}} \dfrac{x \, dx}{ 1 + \sin \, x}$$ =
  • $$\dfrac{\pi}{6} $$
  • $$\pi$$
  • $$\dfrac{\pi}{3}$$
  • Cannot be valued
$$\displaystyle\int  {\dfrac{{\ln \left( {1 + {x}} \right)}}{{1 + {x}}}} dx\,equals$$
  • $$\dfrac {(\ln (1+x))^2}2$$
  • $$ - \pi \ln (1+x)$$
  • $$\frac{\pi }{2}\ln (1+x)$$
  • $$ - \frac{\pi }{2}\ln (1+x)$$
$$\int _{ 0 }^{ 1 }{ \tan ^{ -1 }{ \left( \cfrac { 2x-1 }{ 1+x-{ x }^{ 2 } }  \right)  }  } dx$$ is equal to
  • $$0$$
  • $$1$$
  • $$-1$$
  • None of these
If $$\frac{d}{dx}(\phi (x))=f(x)$$, then $$\int_1^2 {f\left( x \right)} $$ is equal to.
  • $$f(1)-f(2)$$
  • $$\phi(1)-\phi(2)$$
  • $$f(2)-f(1)$$
  • $$\phi(2)-\phi(1)$$
State whether the statement is ture/false.

 $$\displaystyle \int _ { - \pi / 2 } ^ { \pi / 2 } \left( \frac { \sin x } { 1 - \cos x } \right) d x$$=0
  • True
  • False
Evaluate: $$\displaystyle\int _ { 0 } ^ { \pi / 2 } \dfrac { \sin x \cos x } { \cos ^ { 2 } x + 3 \cos x + 2 }dx$$
  • $$\ln\left(\dfrac {5}{3}\right)$$
  • $$\ln\left(\dfrac {4}{3}\right)$$
  • $$\ln\left(\dfrac {1}{3}\right)$$
  • None of these
The value of the integral $$\displaystyle \int_0^{1} {{1 + 2 x}}dx$$ is 
  • $$2$$
  • $$3$$
  • $$4$$
  • $$\frac{ - 1}{4}$$
Solve:
$$\int\limits_0^{\pi /6} {\dfrac{{\cos 2x}}{{{{\left( {\cos x - \sin x} \right)}^2}}}dx} $$
  • $$ - \log \left( {\dfrac{{\sqrt 3 - 1}}{2}} \right)$$
  • $$ - \log \left( {\dfrac{{\sqrt 3 + 1}}{2}} \right)$$
  • $$\log \left( {\dfrac{{\sqrt 3 + 1}}{2}} \right)$$
  • None of these
Solve $$\displaystyle\int\limits_0^{\pi /2} {{{\sin }^4}x{{\cos }^3}xdx} $$ 
  • $$\dfrac {6}{35}$$
  • $$\dfrac {2}{21}$$
  • $$\dfrac {2}{15}$$
  • $$\dfrac {2}{35}$$
Evaluate $$\displaystyle\int \frac{ (\sec \theta)}{(\tan^2 \theta)} d \theta$$
  • $$\dfrac{1}{cos \theta}+c$$
  • $$-\dfrac{1}{sin \theta}+c$$
  • $$\dfrac{1}{tan \theta}+c$$
  • $$\dfrac{1}{cosec \theta}+c$$
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