$\int _ { - \pi } ^ { \pi } | \pi - | \mathrm { x } | | \mathrm { d } x$ is equal to
(1) $\sqrt { 2 } \pi ^ { 2 }$
(2) $2 \pi ^ { 2 }$
(3) $\pi ^ { 2 }$
(4) $\frac { \pi ^ { 2 } } { 2 }$

If for all real triplets $( a , b , c ) , f ( x ) = a + b x + c x ^ { 2 }$; then $\int _ { 0 } ^ { 1 } f ( x ) d x$ is equal to:
(1) $2 \left\{ 3 f ( 1 ) + 2 f \left( \frac { 1 } { 2 } \right) \right\}$
(2) $\frac { 1 } { 2 } \left\{ f ( 1 ) + 3 f \left( \frac { 1 } { 2 } \right) \right\}$
(3) $\frac { 1 } { 3 } \left\{ f ( 0 ) + f \left( \frac { 1 } { 2 } \right) \right\}$
(4) $\frac { 1 } { 6 } \left\{ f ( 0 ) + f ( 1 ) + 4 f \left( \frac { 1 } { 2 } \right) \right\}$

The value of $\int _ { 0 } ^ { 2 \pi } \frac { x \sin ^ { 8 } x } { \sin ^ { 8 } x + \cos ^ { 8 } x } d x$ is equal to:
(1) $2 \pi$
(2) $2 \pi ^ { 2 }$
(3) $\pi ^ { 2 }$
(4) $4 \pi$

The integral $\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 3 } } \tan ^ { 3 } x \cdot \sin ^ { 2 } 3 x \left( 2 \sec ^ { 2 } x \cdot \sin ^ { 2 } 3 x + 3 \tan x \cdot \sin 6 x \right) d x$ is equal to:
(1) $\frac { 7 } { 18 }$
(2) $- \frac { 1 } { 9 }$
(3) $- \frac { 1 } { 18 }$
(4) $\frac { 9 } { 2 }$

The value of $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { 1 } { 1 + e ^ { \sin x } } d x$ is :
(1) $\frac { \pi } { 4 }$
(2) $\pi$
(3) $\frac { \pi } { 2 }$
(4) $\frac { 3 \pi } { 2 }$

If $f ^ { \prime } ( x ) = \tan ^ { - 1 } ( \sec x + \tan x ) , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$ and $f ( 0 ) = 0$, then $f ( 1 )$ is equal to:
(1) $\frac { \pi + 1 } { 4 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { \pi - 1 } { 4 }$
(4) $\frac { \pi + 2 } { 4 }$

If the real part of the complex number $( 1 - \cos \theta + 2i \sin \theta ) ^ { - 1 }$ is $\frac { 1 } { 5 }$ for $\theta \in ( 0 , \pi )$, then the value of the integral $\int _ { 0 } ^ { \theta } \sin x \mathrm {~d} x$ is equal to:
(1) 1
(2) 2
(3) - 1
(4) 0

For $x > 0$, if $f ( x ) = \int _ { 1 } ^ { x } \frac { \log _ { e } t } { ( 1 + t ) } d t$, then $f ( e ) + f \left( \frac { 1 } { e } \right)$ is equal to
(1) 0
(2) $\frac { 1 } { 2 }$
(3) - 1
(4) 1

Consider the integral $I = \int _ { 0 } ^ { 10 } \frac { [ x ] e ^ { [ x ] } } { e ^ { x - 1 } } d x$ where $[ x ]$ denotes the greatest integer less than or equal to $x$. Then the value of $I$ is equal to :
(1) $9 ( e - 1 )$
(2) $45 ( e + 1 )$
(3) $45 ( e - 1 )$
(4) $9 ( e + 1 )$

If $[ x ]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $\int _ { - \pi / 2 } ^ { \pi / 2 } [ [ x ] - \sin x ] \, d x$ is equal to:
(1) $- \pi$
(2) $\pi$
(3) 0
(4) 1

Let $f : R \rightarrow R$ be defined as $f ( x ) = e ^ { - x } \sin x$. If $F : [ 0,1 ] \rightarrow R$ is a differentiable function such that $F ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$, then the value of $\int _ { 0 } ^ { 1 } \left( F ^ { \prime } ( x ) + f ( x ) \right) e ^ { x } d x$ lies in the interval
(1) $\left[ \frac { 327 } { 360 } , \frac { 329 } { 360 } \right]$
(2) $\left[ \frac { 330 } { 360 } , \frac { 331 } { 360 } \right]$
(3) $\left[ \frac { 331 } { 360 } , \frac { 334 } { 360 } \right]$
(4) $\left[ \frac { 335 } { 360 } , \frac { 336 } { 360 } \right]$

If $f : R \rightarrow R$ is given by $f ( x ) = x + 1$, then the value of $\lim _ { n \rightarrow \infty } \frac { 1 } { n } \left[ f ( 0 ) + f \left( \frac { 5 } { n } \right) + f \left( \frac { 10 } { n } \right) + \ldots + f \left( \frac { 5 ( n - 1 ) } { n } \right) \right]$ is:
(1) $\frac { 3 } { 2 }$
(2) $\frac { 5 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 7 } { 2 }$

Let $a$ be a positive real number such that $\int _ { 0 } ^ { a } e ^ { x - [ x ] } d x = 10 e - 9$ where, $[ x ]$ is the greatest integer less than or equal to $x$. Then, $a$ is equal to:
(1) $10 - \log _ { e } ( 1 + e )$
(2) $10 + \log _ { e } 2$
(3) $10 + \log _ { e } ( 1 + e )$
(4) $10 - \log _ { e } 2$

If $f ( x ) = \left\{ \begin{array} { l l } \int _ { 0 } ^ { x } ( 5 + | 1 - t | ) d t , & x > 2 \\ 5 x + 1 , & x \leq 2 \end{array} \right.$, then
(1) $f ( x )$ is not continuous at $x = 2$
(2) $f ( x )$ is everywhere differentiable
(3) $f ( x )$ is continuous but not differentiable at $x = 2$
(4) $f ( x )$ is not differentiable at $x = 1$

The value of $\int _ { - 1 } ^ { 1 } x ^ { 2 } e ^ { \left[ x ^ { 3 } \right] } d x$, where $[ t ]$ denotes the greatest integer $\leq t$, is :
(1) $\frac { e + 1 } { 3 }$
(2) $\frac { e - 1 } { 3 e }$
(3) $\frac { 1 } { 3 e }$
(4) $\frac { e + 1 } { 3 e }$

If the integral $\int _ { 0 } ^ { 10 } \frac { [ \sin 2 \pi x ] } { \mathrm { e } ^ { x - [ x ] } } d x = \alpha e ^ { - 1 } + \beta e ^ { - \frac { 1 } { 2 } } + \gamma$, where $\alpha , \beta , \gamma$ are integers and $[ x ]$ denotes the greatest integer less than or equal to $x$, then the value of $\alpha + \beta + \gamma$ is equal to:
(1) 0
(2) 20
(3) 25
(4) 10

Let $g ( t ) = \int _ { - \pi / 2 } ^ { \pi / 2 } \left( \cos \frac { \pi } { 4 } t + f ( x ) \right) d x$, where $f ( x ) = \log _ { e } \left( x + \sqrt { x ^ { 2 } + 1 } \right) , x \in R$. Then which one of the following is correct?
(1) $g ( 1 ) = g ( 0 )$
(2) $\sqrt { 2 } g ( 1 ) = g ( 0 )$
(3) $g ( 1 ) = \sqrt { 2 } g ( 0 )$
(4) $g ( 1 ) + g ( 0 ) = 0$

Which of the following statement is correct for the function $g ( \alpha )$ for $\alpha \in R$ such that $g ( \alpha ) = \int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 3 } } \frac { \sin ^ { \alpha } x } { \cos ^ { \alpha } x + \sin ^ { \alpha } x } d x$
(1) $g ( \alpha )$ is a strictly increasing function
(2) $g ( \alpha )$ has an inflection point at $\alpha = - \frac { 1 } { 2 }$
(3) $g ( \alpha )$ is a strictly decreasing function
(4) $g ( \alpha )$ is an even function

Let $f : R \rightarrow R$ be defined as $f ( x ) = e ^ { - x } \sin x$. If $F : [ 0,1 ] \rightarrow R$ is a differentiable function such that $F ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$, then the value of $\int _ { 0 } ^ { 1 } \left( F ^ { \prime } ( x ) + f ( x ) \right) e ^ { x } d x$ lies in the interval

Let $f ( x ) = \int _ { 0 } ^ { x } e ^ { t } f ( t ) d t + e ^ { x }$ be a differentiable function for all $x \in R$. Then $f ( x )$ equals:

The value of $\int _ { - \pi / 2 } ^ { \pi / 2 } \left( \frac { 1 + \sin ^ { 2 } x } { 1 + \pi ^ { \sin x } } \right) d x$ is

The value of $\int _ { - 2 } ^ { 2 } \left| 3 x ^ { 2 } - 3 x - 6 \right| d x$ is $\underline{\hspace{1cm}}$.

Let $f : ( 0,2 ) \rightarrow R$ be defined as $f ( x ) = \log _ { 2 } \left( 1 + \tan \left( \frac { \pi x } { 4 } \right) \right)$. Then, $\lim _ { n \rightarrow \infty } \frac { 2 } { n } \left( f \left( \frac { 1 } { n } \right) + f \left( \frac { 2 } { n } \right) + \ldots + f ( 1 ) \right)$ is equal to $\_\_\_\_$.

Let $[ t ]$ denote the greatest integer $\leq \mathrm { t }$. Then the value of $8 \cdot \int _ { - \frac { 1 } { 2 } } ^ { 1 } ( [ 2 x ] + | x | ) \mathrm { d } x$ is

If $x \phi ( x ) = \int _ { 5 } ^ { x } \left( 3 t ^ { 2 } - 2 \phi ^ { \prime } ( t ) \right) d t , x > - 2 , \phi ( 0 ) = 4$, then $\phi ( 2 )$ is