The value of $\alpha$ for which $4 \alpha \int _ { - 1 } ^ { 2 } e ^ { - \alpha | x | } d x = 5$, is
(1) $\log _ { e } 2$
(2) $\log _ { e } \left( \frac { 3 } { 2 } \right)$
(3) $\log _ { e } \sqrt { 2 }$
(4) $\log _ { e } \left( \frac { 4 } { 3 } \right)$

$\int _ { - \pi } ^ { \pi } | \pi - | \mathrm { x } | | \mathrm { d } x$ 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) $\frac { 3 \pi } { 2 }$

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 $P ( x ) = x ^ { 2 } + bx + c$ be a quadratic polynomial with real coefficients such that $\int _ { 0 } ^ { 1 } P ( x ) d x = 1$ and $P ( x )$ leaves remainder 5 when it is divided by $( x - 2 )$. Then the value of $9 ( b + c )$ is equal to:
(1) 9
(2) 15
(3) 7
(4) 11

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) $\frac { 7 } { 2 }$

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

$\lim _ { n \rightarrow \infty } \left[ \frac { 1 } { n } + \frac { n } { ( n + 1 ) ^ { 2 } } + \frac { n } { ( n + 2 ) ^ { 2 } } + \ldots + \frac { n } { ( 2 n - 1 ) ^ { 2 } } \right]$ is equal to
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 1 } { 3 }$
(4) 1

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 $[ t ]$ 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

Let $f : R \rightarrow R$ be a continuous function such that $f ( x ) + f ( x + 1 ) = 2$ for all $x \in R$. If $I _ { 1 } = \int _ { 0 } ^ { 8 } f ( x ) d x$ and $I _ { 2 } = \int _ { - 1 } ^ { 3 } f ( x ) d x$, then the value of $I _ { 1 } + 2 I _ { 2 }$ is equal to $\_\_\_\_$.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined as $f ( x ) = a \sin \left( \frac { \pi [ x ] } { 2 } \right) + [ 2 - x ] , a \in \mathbb { R }$, where $[ t ]$ is the greatest integer less than or equal to $t$. If $\lim _ { x \rightarrow - 1 } f ( x )$ exists, then the value of $\int _ { 0 } ^ { 4 } f ( x ) d x$ is equal to
(1) - 1
(2) - 2
(3) 1
(4) 2

For any real number $x$, let $\lfloor x \rfloor$ denote the largest integer less than or equal to $x$. Let $f$ be a real-valued function defined on the interval $[-10,10]$ by $f(x) = \begin{cases} x - \lfloor x \rfloor, & \text{if } \lfloor x \rfloor \text{ is odd} \\ 1 + \lfloor x \rfloor - x, & \text{if } \lfloor x \rfloor \text{ is even} \end{cases}$ Then, the value of $\frac { \pi ^ { 2 } } { 10 } \int _ { - 10 } ^ { 10 } f(x) \cos(\pi x)\, dx$ is
(1) 4
(2) 2
(3) 1
(4) 0

The value of the integral $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { d x } { \left( 1 + e ^ { x } \right) \left( \sin ^ { 6 } x + \cos ^ { 6 } x \right) }$ is equal to
(1) $2 \pi$
(2) 0
(3) $\pi$
(4) $\frac { \pi } { 2 }$

If $[t]$ denotes the greatest integer $\leq t$, then the value of $\int_0^1 \left[2x - \left|3x^2 - 5x + 2\right| + 1\right] dx$ is
(1) $\frac{\sqrt{37} + \sqrt{13} - 4}{6}$
(2) $\frac{\sqrt{37} - \sqrt{13} - 4}{6}$
(3) $\frac{-\sqrt{37} - \sqrt{13} + 4}{6}$
(4) $\frac{-\sqrt{37} + \sqrt{13} + 4}{6}$

$I = \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \left( \frac { 8 \sin x - \sin 2 x } { x } \right) d x$. Then
(1) $\frac { \pi } { 2 } < I < \frac { 3 \pi } { 4 }$
(2) $\frac { \pi } { 5 } < I < \frac { 5 \pi } { 12 }$
(3) $\frac { 5 \pi } { 12 } < I < \frac { \sqrt { 2 } } { 3 } \pi$
(4) $\frac { 3 \pi } { 4 } < I < \pi$

$\int _ { 0 } ^ { 2 } \left( \left| 2 x ^ { 2 } - 3 x \right| + \left[ x - \frac { 1 } { 2 } \right] \right) d x$, where $[ t ]$ is the greatest integer function, is equal to
(1) $\frac { 7 } { 6 }$
(2) $\frac { 19 } { 12 }$
(3) $\frac { 31 } { 12 }$
(4) $\frac { 3 } { 2 }$

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of the integral $\int_{-3}^{101} \left([\sin(\pi x)] + e^{[\cos(2\pi x)]}\right) dx$ is equal to
(1) $\frac{52(1-e)}{e}$
(2) $\frac{52}{e}$
(3) $\frac{52(2+e)}{e}$
(4) $\frac{104}{e}$