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CBSE Questions for Class 12 Commerce Applied Mathematics Definite Integrals Quiz 4 - MCQExams.com

If f(1x)+x2f(x)=0 for x>0, 
and I=x1/xf(z)dz,12x2 
then I is?
  • f(2)f(1/2)
  • f(1/2)f(2)
  • 0
  • None of these
If \displaystyle I_{1}= \int_{-4}^{-5}e^{\left ( x +5 \right )^{2}}dx and \displaystyle I_{2}= 3\int_{{1}/{3}}^{{2}/{3}}e^{\left ( 3x -2 \right )^{2}}dx 

then I_{1}+I_{2} equals?
  • \displaystyle \frac{1}{3}
  • \displaystyle -\frac{1}{3}
  • 0
  • None of these
If f\left ( a+x \right )= f\left ( x \right ) , then \forall a> 0,  n\epsilon  N the value of \displaystyle \int_{0}^{n a}f\left ( x \right )dx  equals ?
  • \displaystyle \left ( n-1 \right )\int_{0}^{a}f\left ( x \right )dx
  • \displaystyle \left ( 1-n \right )\int_{0}^{a}f\left ( x \right )dx
  • \displaystyle n\int_{0}^{a}f\left ( x \right )dx
  • None of the above
If \displaystyle f\left ( x \right )=\int_{-1}^{1}\frac{\sin x}{1+t^{2}}dt then \displaystyle {f}'\left ( \frac{\pi }{3} \right ) is
  • nonexistent
  • \displaystyle \pi /4
  • \displaystyle \pi \sqrt{3/4}
  • none of these
\displaystyle \int_{0}^{1}\frac{2^{x+1}-3^{x-1}}{6^{x}}dx
  • \displaystyle \frac{4}{3}\log _{3}e-\frac{1}{6}\log _{2}e.
  • \displaystyle -\frac{4}{3}\log _{3}e+\frac{1}{6}\log _{2}e.
  • \displaystyle \frac{4}{3}\log _{3}e-\frac{1}{3}\log _{2}e.
  • \displaystyle -\frac{2}{3}\log _{3}e+\frac{1}{6}\log _{2}e.
\displaystyle \int_{1}^{2}\left ( x+\frac{1}{x} \right )^{3/2}\frac{x^{2}-1}{x^{2}}dx
  • \displaystyle \frac{5}{2}\sqrt{\left ( \frac{5}{2} \right )}+\frac{8}{5}\sqrt{2}
  • \displaystyle \frac{5}{2}\sqrt{\left ( \frac{5}{2} \right )}-\frac{8}{5}\sqrt{2}
  • \displaystyle \sqrt{\left ( \frac{5}{2} \right )}-\frac{8}{5}\sqrt{2}
  • \displaystyle \frac{3}{2}\sqrt{\left ( \frac{3}{2} \right )}-\frac{8}{5}\sqrt{2}
Evaluate the integrals \displaystyle \int_{a}^{b}\frac{\log x}{x}dx
  • \displaystyle \frac{1}{2} \log\left ( ab\right )\cdot \log \frac{b}{a}
  • \displaystyle \frac{1}{4} \log\left ( ab\right )\cdot \log \frac{b}{a}
  • \displaystyle \log\left ( ab\right )\cdot \log \frac{b}{a}
  • \displaystyle -\frac{1}{2} \log\left ( ab\right )\cdot \log \frac{b}{a}
Given \displaystyle \int _{ 1 }^{ 2 }{ { e }^{ { x }^{ 2 } } } dx=a, the value of \displaystyle \int _{ e }^{ { e }^{ 4 } }{ \sqrt { \ln { \left( x \right)  }  } dx } is?
  • { e }^{ 4 }-e
  • { e }^{ 4 }-a
  • 2{ e }^{ 4 }-a
  • 2{ e }^{ 4 }-e-a
\displaystyle \int_{0}^{\pi /2}\frac{dx}{\sin x}equals
  • 0
  • \displaystyle \frac{1}{2}
  • 1
  • 3/2
The value of \displaystyle \int_{0}^{1}\displaystyle \frac{dx}{\left ( x+1 \right )\sqrt{x^{2}+2x}} is
  • \pi /6
  • \pi /3
  • \pi /2
  • \pi
Value of \displaystyle \int_{0}^{\pi /4}\left ( \sqrt{\tan x}-\sqrt{\cot x} \right )\: dx is
  • \sqrt{2}\log \left ( \sqrt{2}-1 \right )
  • \sqrt{2}\log \left ( \sqrt{2}+1 \right )
  • \log \left ( \sqrt{2}+1 \right )
  • \log \left ( \sqrt{2}-1 \right )
The value of \displaystyle \int_{a}^{b}\displaystyle \frac{\log x}{x}\: dx is
  • \log \left ( ab \right )\log \displaystyle \left ( \frac{b}{a} \right )
  • \displaystyle \frac{1}{2}\log \left ( ab \right )\log \displaystyle \left ( \frac{b}{a} \right )
  • \log \left ( a^{2}-b^{2} \right )
  • \left ( a+b \right )\log \left ( a+b \right )
If f\left ( x \right )= A\sin \left ( \dfrac {\pi x}{2} \right )\: +\: B,f{}'\left ( \dfrac 12  \right )= \sqrt{2} and \displaystyle \int_{0}^{1}f\left ( x \right )dx= \displaystyle \frac{2A}{\pi }
then the constants A and B are
  • A=\dfrac {\pi}{2}, B=\dfrac {\pi}{2}
  • A=\dfrac {2}{\pi}, B=3\pi
  • A=0, B=-4\pi
  • A=\dfrac {4}{\pi}, B=0
The value of the integral \displaystyle \int_{0}^{\pi /4}\displaystyle \frac{\sin x+\cos x}{3+\sin 2x}dx is
  • \log 2
  • \log 3
  • \left ( 1/4 \right )\log 3
  • \left ( 1/8 \right )\log 3
The value of \displaystyle \int_{0}^{\pi }\displaystyle \frac{dx}{1-2\alpha \cos x+\alpha ^{2}} is
  • \displaystyle \frac{\pi }{1+\alpha ^{2}} if \alpha > 1
  • \displaystyle \frac{\pi }{\alpha ^{2}-1} if \alpha > 1
  • \displaystyle \frac{\pi }{1+\alpha ^{2}} if \alpha < 1
  • \displaystyle \frac{\pi }{\alpha ^{2}-1} if \alpha < 1
If I= \displaystyle \int_{1}^{2}\displaystyle \frac{dx}{\left ( x+1 \right )\sqrt{x^{2}-1}} then I is equal to
  • 1
  • 1/2
  • 1/\sqrt{2}
  • 1/\sqrt{3}
The value of \displaystyle \int_{0}^{1}\displaystyle \frac{dx}{e^{x}+e^{-x}} is
  • \tan ^{-1}e
  • \tan ^{-1}\left ( e \right )-\pi /4
  • \tan ^{-1}\left ( e \right )-\tan ^{-1}\left ( 1/e \right )
  • \tan ^{-1}\left ( 1/e \right )+\pi /4
The value of \displaystyle \int_{8}^{15}\displaystyle \frac{dx}{\left ( x-3 \right )\sqrt{x+1}} is
  • \log \displaystyle \frac{5}{3}
  • \displaystyle \frac{1}{2}\log \displaystyle \frac{5}{3}
  • 2\log \displaystyle \frac{5}{3}
  • \sqrt{2}\log \left ( \sqrt{2}-1 \right )
If b > a, and \displaystyle  I = \int _{a}^{b} \sqrt{\frac{ x-a}{b-x} }dx, then I equals
  • \displaystyle \frac{\pi}{2} (b -a)
  • \pi (b -a)
  • \pi/2
  • 2\pi (b-a)
Value of \displaystyle \int_{0}^{2a}\dfrac{x^{3/2}}{\sqrt{2a-x}} dx is
  • \displaystyle \frac{3\pi a^{2}}{2}
  • \pi a^{3}
  • \sqrt{2}\pi a^{3}
  • 2\pi a^{3}
Value of \displaystyle \int_{0}^{25}\displaystyle \frac{1}{\sqrt{4+\sqrt{x}}}\: dx is
  • 2\left ( \sqrt{29}-1 \right )
  • 2\left ( \sqrt{29}-5 \right )
  • 3 \sqrt{29}-1
  • none of these
\displaystyle \int_{0}^{\infty }f\left ( x+\frac{1}{x} \right )\frac{\ln x}{x}dx
  • is equal to zero
  • is equal to one
  • is equal to \displaystyle \frac{1}{2}
  • can not be evaluated
If\displaystyle \int_{0}^{\pi /3}\frac{\cos }{3+4\sin x}dx=K\log \frac{\left ( 3+2\sqrt{3} \right )}{3} then K is
  • \displaystyle \frac{1}{2}
  • \displaystyle \frac{1}{3}
  • \displaystyle \frac{1}{4}
  • \displaystyle \frac{1}{8}
Suppose that F(x) is an antiderivative of f(x)\displaystyle =\frac{\sin x}{x},x> 0 then \displaystyle \int_{1}^{3}\frac{\sin 2x}{x} can be expressed as
  • F(6) - F(2)
  • \displaystyle \frac{1}{2}\left ( F\left ( 6 \right )-F\left ( 2 \right ) \right )
  • \displaystyle \frac{1}{2}\left ( F\left ( 3 \right )-F\left ( 1 \right ) \right )
  • \displaystyle 2\left ( F\left ( 6 \right )-F\left ( 2 \right ) \right )
If 0 < \alpha < 1 and \displaystyle I=\int _{-1}^{1} \frac{dx}{\sqrt{1-2\alpha x+\alpha^{2}}} then I equals
  • 1/\alpha
  • 2/\alpha
  • 3/\alpha
  • none of these
\displaystyle \int_{1/2}^{2}\frac{1}{x}\sin \left ( x-\frac{1}{x} \right )dx has the value equal to 
  • 0
  • \displaystyle \frac{3}{4}
  • \displaystyle \frac{5}{4}
  • 2
The value of the definite integral \displaystyle \int_{1}^{\infty}\left ( e^{x+1}+e^{3-x} \right )^{-1}dx is
  • \displaystyle \frac{\pi }{4e^{2}}
  • \displaystyle \frac{\pi }{4e}
  • \displaystyle \frac{1}{e^{2}}\left ( \frac{\pi }{2}-\tan ^{-1}\frac{1}{e} \right )
  • \displaystyle \frac{\pi }{2e^{2}}
\displaystyle \int ^{\displaystyle \frac{3 \pi}{10}}_{\displaystyle \frac{\pi}{5}} \frac{sin x}{sin x + cos x}dx is equal to 
  • \pi
  • \displaystyle \frac{\pi}{2}
  • \displaystyle \frac{\pi}{4}
  • \displaystyle \frac{\pi}{20}
If \displaystyle f\left ( x \right )=\int_{1}^{x}\frac{\ln t}{1+t}dt where x > 0, then the value(s) of x satisfying the equation, f(x) +f(1/x)=2 is
  • 2
  • e
  • \displaystyle e^{-2}
  • \displaystyle e^{2}
Choose a function f(x) such that it is integrable over every interval on the real line
  • f(x) = [x]
  • f(x)=x|x|
  • f(x)=[sinx]
  • f(x)=\dfrac{|x-1|}{x-1}
\displaystyle \int_{0}^{\infty }\frac{x}{\left ( 1+x \right )\left ( 1+x^{2} \right )}dx
  • \displaystyle \frac{\pi }{4}
  • \displaystyle \frac{\pi }{2}
  • is sme as \displaystyle \int_{0}^{\infty }\frac{dx}{\left ( 1+x \right )\left ( 1+x^{2} \right )}
  • cannot be evaluated
State true or false:
The average value of the function f(x) = sin^2xcos^3x on the interval [ -\pi ,\pi ] is 0.
  • True
  • False
I_{1} is equal to
  • \displaystyle \frac {2}{3} \int_{0}^{\pi/2}(sin^{2}\theta)(cos\theta)^{-1/3}d\theta
  • \displaystyle \frac {3}{2} \int_{0}^{\pi/2}(sin^{2}\theta)(cos\theta)^{-1/3}d\theta
  • \displaystyle \frac {2}{3} \int_{0}^{\pi/2}(sin\theta)^{2/3}(cos\theta)^{-1/3}d\theta
  • \displaystyle \frac {3}{2} \int_{0}^{\pi/2}(sin\theta)^{2/3}(cos\theta)^{-1/3}d\theta
Evaluate \displaystyle \int_{0}^{\pi /2} \frac{dx}{2+\sin 2x}
  • \displaystyle \frac{2\pi }{{3}}
  • \displaystyle \frac{\pi }{{3}}
  • \displaystyle \frac{2\pi }{{5}}
  • None of these
  • Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
  • Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
  • Assertion is correct but Reason is incorrect
  • Both Assertion and Reason are incorrect
\displaystyle \int_{\tfrac{1}{\sqrt{3}}}^{0}\dfrac{dx}{\left ( 2x^{2}+1 \right )\sqrt{x^{2}+1}}
  • \displaystyle - \tan^{-1} \dfrac{1}{2}
  • \displaystyle \tan^{-1} 1
  • \displaystyle - \tan^{-1} \dfrac{1}{3}
  • \displaystyle \tan^{-1} \dfrac{1}{\sqrt 2}
The value of definite integral \displaystyle \int_{\infty }^{0}\frac{Ze^{-z}}{\sqrt{1-e^{-2z}}}dz
  • \displaystyle -\frac{\pi }{2}ln2
  • \displaystyle \frac{\pi }{2}ln2
  • \displaystyle -\pi ln2
  • \displaystyle \pi ln\frac{1}{\sqrt{2}}
The derivative of F(x)=\displaystyle\int^{x^3}_{x^2}\frac{dt}{log t}(x >0) is 
  • \displaystyle\frac{1}{3log x}
  • \displaystyle\frac{1}{3log x}-\frac{1}{2log x}
  • (log x)^{-1}x(x-1)
  • \displaystyle\frac{3x^2}{log x}
If \displaystyle \int _{ 0 }^{ 1 }{ { e }^{ { x }^{ 2 } }\left( x-\alpha  \right)  } dx=0, then
  • 1<\alpha <2
  • \alpha <2
  • 0<\alpha<1
  • \alpha=0
Trapezoidal rule for evaluation of \displaystyle\int _{ a }^{ b }{ f(x)dx } requires the interval (a,b) to be divided into
  • 2n sub-intervals of equal width
  • any number of sub intervals of not equal width
  • any number of sub intervals of equal width
  • 3n sub-intervals of equal width
\displaystyle \int _{ 0 }^{ a }{ \frac { dx }{ a+\sqrt { { a }^{ 2 }-{ x }^{ 2 } }  }  } is equal to
  • \displaystyle \frac { \pi  }{ 2 } +1
  • \displaystyle \frac { \pi  }{ 2 } -1
  • \displaystyle 1-\frac { \pi  }{ 2 }
  • none of these
The value of the integral \int_0^{\dfrac{\pi}{2}}\,sin^5x\,dx is
  • \dfrac{4}{15}
  • \dfrac{8}{5}
  • \dfrac{8}{15}
  • \dfrac{4}{5}
Using Trapezoidal rule and following table \int_0^8 {f(x)}dx is equal to 
x02468
f(x)25101726
  • 184
  • 92
  • 46
  • -36
\displaystyle \int_1^{\sqrt 3}\frac {dx}{1+x^2} equals
  • \dfrac {\pi}{3}
  • \dfrac {2\pi}{3}
  • \dfrac {\pi}{6}
  • \dfrac {\pi}{12}
\displaystyle\int _{ 1 }^{ 3 }{ \dfrac { \cos { \left( \log { x }  \right)  }  }{ x } dx } is equal to
  • 1
  • \cos { \left( \log { 3 } \right) }
  • \sin { \left( \log { 3 } \right) }
  • \dfrac { \pi }{ 4 }
The value of I=\displaystyle \int _{ 0 }^{ \frac { \pi  }{ 4 }  }{ \left( \tan ^{ n+1 }{ x }  \right) dx } +\frac { 1 }{ 2 } \int _{ 0 }^{ \frac { \pi  }{ 2 }  }{ \left( \tan ^{ n+1 }{ \left( \frac { x }{ 2 }  \right)  }  \right) dx } is
  • \cfrac{1}{n}
  • \cfrac{n+2}{2n+1}
  • \cfrac{2n-1}{n}
  • \cfrac{2n-3}{3n-2}
What is \displaystyle \int _{ 0 }^{ 1 }{ \cfrac { \tan ^{ -1 }{ x }  }{ 1+{ x }^{ 2 } }  } dx equal to?
  • \cfrac { { \pi }^{ 2 } }{ 8 }
  • \cfrac { { \pi }^{ 2 } }{ 32 }
  • \cfrac { \pi }{ 4 }
  • \cfrac { \pi }{ 8 }
What is \displaystyle\int _{ 1 }^{ 2 }{ \ln { x } dx } equal to?
  • \ln { 2 }
  • 1
  • \ln { \left( \dfrac { 4 }{ e } \right) }
  • \ln { \left( \dfrac { e }{ 4 } \right) }
If \displaystyle \int_{0}^{b} \displaystyle \frac{dx}{1\, +\, x^2}\, =\, \displaystyle \int_{b}^{\infty} \displaystyle \frac{dx}{1\, +\, x^2}, then b =
  • \tan ^{-1}\, \left (\displaystyle \frac{1}{3} \right )
  • \displaystyle \frac{\sqrt 3}{2}
  • \sqrt 3
  • 1
Select and write the most appropriate answer from the given alternatives for question :
If \displaystyle \int^k_0 4x^3dx=16, then the value of k is _____.
  • 1
  • 2
  • 3
  • 4
0:0:1


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Practice Class 12 Commerce Applied Mathematics Quiz Questions and Answers