Test Details

Integrals

Intregration

Jan 24, 1:00 PM

120 minutes

Jan 24, 3:00 PM

100.0 marks

MCQ

50 questions

Navneet

heat engine

Class Details

Physics

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Question Text

Question 1 :
What is $\displaystyle \int \dfrac{x^4 - 1}{x^2 \sqrt{x^4 + x^2 + 1}} dx$ equal to?

Question 3 :
 Find Integrals of given function: $\int_{}^{} {\tan \theta } {\tan ^2}\theta {\sec ^2}\theta d\theta $

Question 4 :
If $g\left( x \right) =\int { { x }^{ x }\log _{ e }{ (ex)dx }  } $ then  $g\left( \pi \right) $ equals

Question 5 :
$\int _{ 0 }^{ \pi /4 }{ \cfrac { \sec ^{ 2 }{ x } }{ \left( 1+\tan { x } \right) \left( 2+\tan { x } \right) } dx } $ equals:

Question 6 :
If $\int { \cfrac { 1-{ \left( \cot { x } \right) }^{ 2010 } }{ \tan { x } +{ \left( \cot { x } \right) }^{ 2011 } } dx } =\cfrac { 1 }{ k } \log _{ e }{ \left| { \left( \sin { x } \right) }^{ k }+{ \left( \cos { x } \right) }^{ k } \right| } +C$, then $k$ is equal to

Question 9 :
The value of $\int_{1}^{e} \dfrac{1+x^{2} \ln x}{x+x^{2} \ln x} d x$ is

Question 13 :
$f(x), g(x)$ are two differentiable function on $[0, 2]$ such that ${f}''\left ( x \right )-{g}''\left ( x \right )=0$ and ${f}'\left ( 1 \right )=4=2{g}'\left ( 1 \right )$ and $f\left ( 2 \right )=3g\left ( 2 \right )=9$ then $\left [ f\left ( x \right )-g\left ( x \right ) \right ]$ at $\displaystyle x=\dfrac{3}{2}$ is

Question 14 :
The value of $\displaystyle\int { \dfrac { dx }{ \sqrt { 2x-{ x }^{ 2 } } } } $ is

Question 18 :
The integral $\displaystyle \int { \cfrac { x+2 }{ \left( { x }^{ 2 }+3x+3 \right) \sqrt { x+1 }  }  }dx $ is equl to

Question 19 :
If $\int \sqrt 2\sqrt{1+\sin x}dx = -4 \cos(ax+b)+c$, then the value of a,b are:

Question 21 :
The value of the definite integral $\overset { { a }_{ 1 } }{ \underset { { a }_{ 2 } }{ \int { \frac { d\theta  }{ 1+tan\theta  }  }  }  } =\frac { 501\pi  }{ K } $ where $\ a _{ 2 }=\quad \frac { 1003\pi  }{ 2008 } $ and ${ \ a  }_{ 1 }=\frac { \pi  }{ 2008 } $ The value of K equalls

Question 22 :
The value of $\displaystyle\int { \cfrac { \sin { x } +\cos { x }  }{ 3+\sin { 2x }  }  } dx$ is

Question 23 :
$\displaystyle\int { \cfrac { 1 }{ 7 } \sin { \left( \cfrac { x }{ 7 } +10 \right)  } dx } $ is equal to

Question 25 :
$\displaystyle\int { \dfrac { dx }{ x+\sqrt { x } } } $ is equal to

Question 26 :
If $g(1)=g(2)$ then the value of  $\int _{ 1 }^{ 2 }{ { \left[ f\{ g(x)\}  \right]  }^{ -1 } } f'\{ g(x)\} g'(x)dx\quad is$

Question 28 :
Let $f\left( x \right) $ be a polynomial of degree three satisfying $f\left( 0 \right) =-1$ and $f(1)=0$. Also, $0$ is a stationary point of $f(x)$. If $f(x)$ does not have an extremum at $x=0$, then $\displaystyle\int { \frac { f\left( x \right) }{ { x }^{ 3 }-1 } dx } $ is equal to

Question 31 :
$\displaystyle \int {\frac {1}{x \sqrt{x^2 - 1} } } dx $ is equal to :

Question 35 :
What is $\int { (x^2 + 1)^{\frac{5}{2}}xdx}$ equal to ? Where c is a constant of integration.

Question 38 :
$\displaystyle \int \cos \left \{ 2\tan ^{-1}\sqrt{\frac{1-x}{1+x}} \right \}dx$ is equal to

Question 39 :
If $\int \frac{x\, cos\,  \alpha+1 }{(x^2+2x\, cos\,  \alpha+1)^{3/2}}$ $dx= \frac{x}{\sqrt{f(x) + g(x)cos\, \alpha }}+c$ then (more than one option is correct)

Question 42 :
If $\displaystyle\int { { x }^{ \frac { 13 }{ 2 } }.{ \left( 1+{ x }^{ \frac { 5 }{ 2 } } \right) }^{ \frac { 1 }{ 2 } }dx } =A{ \left( 1+{ x }^{ \frac { 5 }{ 2 } } \right) }^{ \frac { 7 }{ 2 } }+B{ \left( 1+{ x }^{ \frac { 5 }{ 2 } } \right) }^{ \frac { 5 }{ 2 } }+C{ \left( 1+{ x }^{ \frac { 5 }{ 2 } } \right) }^{ \frac { 3 }{ 2 } }$, then

Question 44 :
If $\int_{1}^{2} e^{x^{2}} d x=a,$ then $\int_{e}^{e^{4}} \sqrt{\ln x} d x$ is equal to

Question 47 :
Integrate the following function with respect to x$\displaystyle \int \left ( 5x^{2}+3x-2 \right )dx$

Question 48 :
$\displaystyle\int { \dfrac { dx }{ x+\sqrt { x } } } $ equals

Question 49 :
If $\int \sin x d (\sec  x) = f(x) - g(x) + c$, then

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