CBSE Questions for Class 12 Commerce Maths Integrals Quiz 5 - MCQExams.com

$$y= \int \sqrt{1+\sin2x} dx; y $$ is equal to-
  • $$\sin x - \cos x+c$$
  • $$\sin x+ \cos x +c$$
  • $$ \cos x - \sin x +c$$
  • None of these
The value of $$\displaystyle \int _0^{\pi/2} \sin x \cos x dx $$
  • $$\dfrac 12$$
  • $$\dfrac 34$$
  • $$2$$
  • $$None\ of\ these$$
$$\int _{ 0 }^{ \pi  }{ x\ln { \left( \sin { x }  \right)  }  } dx=$$
  • $$\dfrac { \pi }{ 2 } \ln { 2 } $$
  • $$\dfrac { -\pi^{2} }{ 2 } \ln { 2 } $$
  • $$\dfrac { -\pi }{ 2 } \ln { 2 } $$
  • $$-2p\ \ln { 2 } $$
$$\displaystyle\int^{\pi/4}_0\sec^4xdx$$.
  • $$\dfrac{3}{4}$$
  • $$\dfrac{-3}{4}$$
  • $$\dfrac{4}{3}$$
  • $$\dfrac{5}{4}$$
$$\int_{0}^{2\pi}{ln(1+\cos{x})}dx=$$
  • $$\pi\ln{2}$$
  • $$-\pi\ln{2}$$
  • $$-2\pi\ln{2}$$
  • $$2\pi\ln{2}$$
Evaluate $$\displaystyle \int_\cfrac{\pi}{4}^\cfrac{\pi}{2}(\sqrt{\tan x}+\sqrt{\cot x})dx=$$
  • $$\dfrac{\pi}{2\sqrt{2}}$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{\sqrt{2}}$$
  • $$\dfrac{\pi}{3}$$
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 
  • $$4(x - \tan^{-1} x), \, \ \text{if} \, x < 0$$
  • $$0 \, if \, x > 0$$
  • $$\ln ( 1 + x^3) \, \ \text{if} \, x > 0 $$
  • $$4(x + \tan^{-1} x ) \, \text{if} \, x < 0$$
$$\displaystyle \int _0^4\dfrac{2x+3}{x^2+3x+2}dx$$
  • $$\log 3$$
  • $$\log 5$$
  • $$\log 15$$
  • $$\log 2$$
Evaluate $$\displaystyle\int{\dfrac{x}{\sqrt{{{x}^{2}}+2}}dx}$$
  • $$I=\sqrt{{{x}^{2}}-2}+C$$
  • $$I=\sqrt{{{x}^{2}}+2}+C$$
  • $$I=\sqrt{{{x}^{3}}+2}+C$$
  • $$I=\sqrt{{{x}^{3}}-2}+C$$
$$\displaystyle \int \sqrt{\dfrac{a+x}{a-x}}dx$$ is equal to-
  • $$a\sin^{-1} (x/a)-\sqrt{a^{2}-x^{2}}+c$$
  • $$a\cos^{-1} (x/a)-\sqrt{a^{2}-x^{2}}+c$$
  • $$a\sin^{-1} (x/a)-\sqrt{a^{2}+x^{2}}+c$$
  • $$a\cos^{-1} (x/a)-\sqrt{a^{2}+x^{2}}+c$$
$$\int ( 1+3x+3x^2+4x^3+........)dx (|x| <1)$$-
  • $$(1+x)^{-1}+c$$
  • $$(1-x)^{-1}+c$$
  • $$(1+x)^{-2}+c$$
  • None of these
The angle between the tangent lines to the graph of the function $$f(x) =\int_\limits{2}^x (2t -5)dt$$ at the point where the graph cuts the $$x$$-axis is
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{\pi}{2}$$
$$ \int { P\left( x \right) { e }^{ kx }dx=Q\left( x \right) { e }^{ 4x }+C }$$, where $$P(x)$$ is polynomial of degree $$n$$ and $$Q(x)$$ is polynomial of degree $$7$$. Then the value of $$ n+7+k+\lim _{ x\rightarrow \infty  }{ \dfrac { P\left( x \right)  }{ Q\left( x \right)  }  }$$ is:
  • $$18$$
  • $$19$$
  • $$20$$
  • $$22$$
For $$x > 0$$, let $$f(x)=\displaystyle \int_{1}^{x}\dfrac {\log t}{1+t} \ dt$$  then, the value of  $$f(x)+f(1/x)$$ will be
  • $$\dfrac {1}{4}\log x^{2}$$
  • $$\log x$$
  • $$\dfrac {1}{4}(\log x)^{2}$$
  • $$\dfrac {1}{2}(\log x)^{2}$$
Evaluate using limit of sum:
$$\displaystyle \int_{1}^{3} {(x+1)^2}dx$$
  • $$26$$
  • $$28$$
  • $$30$$
  • $$32$$
$$ \displaystyle \int { \dfrac { { 2 }^{ x } }{ \sqrt { 1-{ 4 }^{ x } }  } dx=K\sin ^{ -1 }{ \left( { 2 }^{ x } \right) +C }  }$$, then the value of $$K$$ is equal to
  • $$\ell n 2$$
  • $$\dfrac{1}{2}\ell 2$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{\ell n 2}$$
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
$$\displaystyle \overset{\ln \pi}{\underset{\ln\pi - \ln2}{\int}} \dfrac{e^x}{1 - \cos \left(\tfrac{2}{3}e^x\right)}dx$$ is equal to
  • $$\sqrt{3}$$
  • $$-\sqrt{3}$$
  • $$\dfrac{1}{\sqrt{3}}$$
  • $$-\dfrac{1}{\sqrt{3}}$$
If $$\displaystyle \int_{0}^{a} \dfrac{dx}{\sqrt{x+a}+\sqrt{x}}=\displaystyle \int_{0}^{\pi/8} \dfrac{2 \tan \theta}{\sin 2 \theta} d\theta$$, then value of $$a$$ is equal to $$(a > 0)$$
  • $$\dfrac{3}{4}$$
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{3\pi}{4}$$
  • $$\dfrac{9}{16}$$
$$\int {{e^x}\left[ {{\mathop{\rm tanx}\nolimits}  - log\left( {\cos x} \right)} \right]} dx = $$
  • $${e^x}\log \left( {\sec x} \right) + c$$
  • $${e^x}\log \left( {co\sec x} \right) + c$$
  • $${e^x}\log \left( {\cos x} \right) + c$$
  • $${e^x}\log \left( {\sin x} \right) + c$$
If $$\displaystyle \int \dfrac{dx}{\sqrt{\sin^3 x \cos^5 x}} = a \sqrt{\cot x } + b \sqrt {\tan^3x} + c$$ where c is an arbitrary constant of integration then the values of $$'a'$$ and $$'b'$$ are respectively :
  • $$-2 $$ & $$\dfrac{2}{3}$$
  • $$2 $$ & $$-\dfrac{2}{3}$$
  • $$2 $$ & $$\dfrac{2}{3}$$
  • 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:
$$\displaystyle\int_{0}^{\pi/2}\dfrac {\sin x-\cos x}{1+\sin x\cos x}dx$$ 
  • $$0$$
  • $$1$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{4}$$
Evaluate $$\displaystyle\int^{\pi}_0\dfrac{x}{1+\sin x}dx$$.
  • $$x\tan x - ln|\cos x| - x\sec x -ln|\sec x - \tan x| + C$$
  • $$x\tan x + ln|\cos x| - x\sec x -ln|\sec x - \tan x| + C$$
  • $$\tan x + ln|\cos x| - x\sec x -ln|\sec x - \tan x| + C$$
  • 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
$$\displaystyle \int (x^2-x+5)\, dx$$
  • $$\dfrac {x^3}3-\dfrac{x^2}2+5x+c$$
  • $$\dfrac {x^3}3+\dfrac{x^2}2+5x+c$$
  • $$\dfrac {x^2}2-\dfrac{x}2+5x+c$$
  • $$\dfrac {x^4}4-\dfrac{x^4}3+5$$
$$\displaystyle \int {{x^3}{e^{{x^2}}}dx = } $$
  • $$\dfrac{1}{2}\left( {{x^2} + 1} \right){e^{{x^2}}} + c$$
  • $$\left( {{x^2} + 1} \right){e^{{x^2}}} + c$$
  • $$\dfrac{1}{2}\left( {{x^2} - 1} \right){e^{{x^2}}} + c$$
  • $$\left( {{x^2} - 1} \right){e^{{x^2}}} + c$$
$$\int \sqrt {1 + \sin x}dx =$$
  • $$\dfrac {1}{2}\left (\sin \dfrac {x}{2} + \cos \dfrac {x}{2}\right ) + c$$
  • $$\dfrac {1}{2}\left (\sin \dfrac {x}{2} - \cos \dfrac {x}{2}\right ) + c$$
  • $$2\sqrt {1 + \sin x} + c$$
  • $$-2\sqrt {1 - \sin x} + c$$
$$\displaystyle\int x^2e^{x^3}\cos \left(e^{x^3}\right)dx$$ is equal to?
  • $$\sin\left(e^{x^3}\right)+C$$
  • $$3\sin \left(e^{x^3}\right)+C$$
  • $$\dfrac{1}{3}\sin\left(e^{x^3}\right)+C$$
  • $$e^x\sin\left(e^{x^3}\right)+C$$
$$\displaystyle \underset{0}{\overset{\pi}{\int}} \dfrac{x \, dx}{1 + \sin \, x} = $$
  • $$\dfrac{\pi}{6}$$
  • $$\pi$$
  • $$\dfrac{\pi}{3}$$
  • none of these
The integral of $$\displaystyle\int e^{\sin x}(x\cos x-\sec x\tan x)dx$$ is?
  • $$se^{\sin x}-e^{\sin x}\sec x+c$$
  • $$(x+\sec x)e^{\sin x}+c$$
  • $$e^{\sin x}\cos x+c$$
  • $$e^{\sin x}(\cos x-\sec x)+c$$
$$\int\dfrac{e^x+e^{-x}+(e^x-e^{-x})sin x}{1+cos x}dx=$$
  • $$(e^x+e^{-x}) tan (x/2)+C$$
  • $$(e^x-e^{-x}) cot (x/2)+C$$
  • $$(e^x-e^{-x}) tan (x/2)+C$$
  • $$(e^x-e^{-x}) cosec (x/2)+C$$
What is the value of $$\int_{0}^{a}\dfrac{x-a}{x+a}\ dx$$?
  • $$a+2a\log 2$$
  • $$a-2a\log 2$$
  • $$2a\log 2-a$$
  • $$2a\log 2$$
$$\int \dfrac {dx}{(x^{2} + 4x + 5)^{2}}$$ is equal to
  • $$\dfrac {1}{2}\left [\tan^{-1}(x + 1) + \dfrac {x + 2}{x^{2} + 4x + 5}\right ] + c$$
  • $$\dfrac {1}{2}\left [\tan^{-1}(x + 2) - \dfrac {x + 2}{x^{2} + 4x + 5}\right ] + c$$
  • $$\dfrac {1}{2}\left [\tan^{-1}(x + 1) - \dfrac {x + 2}{x^{2} + 4x + 5}\right ] + c$$
  • $$\dfrac {1}{2}\left [\tan^{-1}(x + 2) + \dfrac {x + 2}{x^{2} + 4x + 5}\right ] + c$$
$$\displaystyle\int  {\dfrac{{\ln \left( {1 + {x}} \right)}}{{1 + {x}}}} dx\,equals$$
  • $$\dfrac {(\ln (1+x))^2}2$$
  • $$ - \pi \ln (1+x)$$
  • $$\frac{\pi }{2}\ln (1+x)$$
  • $$ - \frac{\pi }{2}\ln (1+x)$$
The value of $$\int\limits_0^{\pi /2} {\dfrac{{x\sin x\cos x}}{{{{\sin }^4}x + {{\cos }^4}x}}dx\,is} $$ is 
  • $$\dfrac{{{\pi ^2}}}{4}$$
  • $$\dfrac{{{\pi ^2}}}{8}$$
  • $$\dfrac{{{\pi ^2}}}{{16}}$$
  • $$\dfrac{{3{\pi ^2}}}{{16}}$$
Find the value: 
$$I=\int_{1}^{2}{\frac{xdx}{(x+1)(x+2)}}$$
  • $$\log \dfrac{32}{27}$$
  • $$\log \dfrac{27}{20}$$
  • $$\log \dfrac{27}{16}$$
  • $$\log \dfrac{32}{9}$$
$$\displaystyle \int_{1/e}^{e}{|\ln x|dx}$$ equals
  • $$e^{-1}-1$$
  • $$2\left (1-\dfrac {1}{e}\right)$$
  • $$1-\dfrac {1}{e}$$
  • $$e-1$$
If $$y=(x+\sqrt{x^{2}-a^{2}})^{n}$$ then $$(x^{2}-a^{2})(\dfrac{dy}{dx})^{2}$$=
  • $$n^{2}y$$
  • $$-n^{2}y$$
  • $$ny^{2}$$
  • $$n^{2}y^{2}$$
$$\int _{ 0 }^{ \pi /4 }{ x.\sec ^{ 2 }{ x } dx=? }$$
  • $$\frac {\pi}{4}+\log {\sqrt {2}}$$
  • $$\frac {\pi}{4}-\log {\sqrt {2}}$$
  • $$1+\log {\sqrt {2}}$$
  • $$1-\frac{1}{2}\log {2}$$
$$\int\limits_0^\pi  {\frac{{{x^2}{{\cos }^4}x\sin x}}{{2\pi x - {\pi ^2}}}dx}  = $$
  • $$\frac{2}{5}$$
  • $$\frac{1}{5}$$
  • $$\frac{{2\pi }}{5}$$
  • $$\frac{\pi }{5}$$
$$\displaystyle \int\limits_0^\frac \pi 2 x\sqrt {1-x^2} dx = $$
  • $$\dfrac{\pi }{4}$$
  • $$\dfrac{\pi }{3}$$
  • $$\dfrac{\pi }{6}$$
  • $$\dfrac{\pi}{8}$$
$$\int\limits_0^{\dfrac{ \pi }{4}} {\sin x.{{\sec }^3}xdx} $$
  • $$\dfrac{1}{2}$$
  • $$\sqrt 2 $$
  • $$2$$
  • $$\dfrac{1}{3}$$
$$\int \dfrac {\sin^{-1}x-\cos^{-1}x}{\sin^{-1}x+\cos^{-1}x}dx=$$
  • $$\dfrac {4}{\pi}[x \sin^{-1}+\sqrt {1-x^{2}}]-x+c$$
  • $$\log [\sin^{-1}+\sqrt {1-x^{2}}]-x+c$$
  • $$\dfrac {4}{\pi}[x \sin^{-1}+\sqrt {1-x^{2}}]+c$$
  • $$\dfrac {2}{\pi}[x \sin^{-1}x-x \cos^{-1}x+2\sqrt {1-x^{2}}]+c$$
$$\int \dfrac {x+\sin }{1+\cos x}dx=$$
  • $$x\ \tan \dfrac {x}{2}+c$$
  • $$x\ \cot \dfrac {x}{2}$$
  • $$x\ \sin\dfrac {x}{2}+c$$
  • $$x\ \cos \dfrac {x}{2}$$
Solve $$\displaystyle \int { x\sin ^{ 2 }{ x }  } dx$$ 
  • $$\dfrac{(x-1)}{2}(x-\dfrac{cos2x}{2})+C$$ 
  • $$\dfrac{(x-1)}{2}(x-\dfrac{sin2x}{2})+C$$ 
  • $$\dfrac{(x+1)}{2}(x-\dfrac{sin2x}{2})+C$$ 
  • $$None\ of\ these$$
Solve $$\int^{3}_{2}\dfrac {x}{x^{2}-1}dx$$
  • $$\dfrac{1}{2}ln(\dfrac{5}{3})$$
  • $$\dfrac{1}{2}ln(\dfrac{8}{3})$$
  • $$\dfrac{1}{2}ln(\dfrac{4}{3})$$
  • None of these
$$\int _{ 0 }^{ \pi /6 }{ \dfrac { \sin { x }  }{ \cos ^{ 3 }{ x }  } dx } =$$
  • $$2/3$$
  • $$1/3$$
  • $$2$$
  • $$1/6$$
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