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) $9 ( e + 1 )$

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) $10 - \log _ { e } 2$

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) $\frac { e + 1 } { 3 e }$

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

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

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 $[ t ]$ 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 function defined by : $f ( x ) = \left\{ \begin{array} { c c } \max _ { t \leq x } \left\{ t ^ { 3 } - 3 t \right\} ; & x \leq 2 \\ x ^ { 2 } + 2 x - 6 ; & 2 < x < 3 \\ { [ x - 3 ] + 9 ; } & 3 \leq x \leq 5 \\ 2 x + 1 ; & x > 5 \end{array} \right.$ Where $[ t ]$ is the greatest integer less than or equal to $t$. Let $m$ be the number of points where $f$ is not differentiable and $I = \int _ { - 2 } ^ { 2 } f ( x ) d x$. Then the ordered pair ( $m , I$ ) is equal to
(1) $\left( 3 , \frac { 27 } { 4 } \right)$
(2) $\left( 3 , \frac { 23 } { 4 } \right)$
(3) $\left( 4 , \frac { 27 } { 4 } \right)$
(4) $\left( 4 , \frac { 23 } { 4 } \right)$

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) $\frac{104}{e}$

$\int _ { 0 } ^ { 20 \pi } ( | \sin x | + | \cos x | ) ^ { 2 } \, d x$ is equal to:
(1) $10 \pi + 4$
(2) $10 \pi + 2$
(3) $20 \pi - 2$
(4) $20 \pi + 2$

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) $\frac{-\sqrt{37} + \sqrt{13} + 4}{6}$

The integral $\int _ { 0 } ^ { 1 } \frac { 1 } { 7 ^ { \left[ \frac { 1 } { x } \right] } } d x$, where $[ \cdot ]$ denotes the greatest integer function, is equal to
(1) $1 - 6 \ln \left( \frac { 6 } { 7 } \right)$
(2) $1 + 6 \ln \left( \frac { 6 } { 7 } \right)$
(3) $1 - 7 \ln \left( \frac { 6 } { 7 } \right)$
(4) $1 + 7 \ln \left( \frac { 6 } { 7 } \right)$

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

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

Let $f ( x ) = 2 + | x | - | x - 1 | + | x + 1 | , x \in R$. Consider $( S 1 ) : f ^ { \prime } \left( - \frac { 3 } { 2 } \right) + f ^ { \prime } \left( - \frac { 1 } { 2 } \right) + f ^ { \prime } \left( \frac { 1 } { 2 } \right) + f ^ { \prime } \left( \frac { 3 } { 2 } \right) = 2$ $( S 2 ) : \int _ { - 2 } ^ { 2 } f ( x ) d x = 12$ Then,
(1) both ( $S 1$ ) and ( $S 2$ ) are correct
(2) both $( S 1 )$ and $( S 2 )$ are wrong
(3) only ( $S 1$ ) is correct
(4) only ( $S 2$ ) is correct

$\int _ { 0 } ^ { 2 } \left( \left| 2 x ^ { 2 } - 3 x \right| + \left[ x - \frac { 1 } { 2 } \right] \right) d x$, 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 $f ( x ) = \min \{ [ x - 1 ] , [ x - 2 ] , \ldots , [ x - 10 ] \}$ where $[ t ]$ denotes the greatest integer $\leq t$. Then $\int _ { 0 } ^ { 10 } f ( x ) d x + \int _ { 0 } ^ { 10 } ( f ( x ) ) ^ { 2 } d x + \int _ { 0 } ^ { 10 } | f ( x ) | d x$ is equal to $\_\_\_\_$ .

Let $[ \mathrm { x } ]$ denote the greatest integer $\leq \mathrm { x }$. Consider the function $\mathrm { f } ( \mathrm { x } ) = \max \left\{ \mathrm { x } ^ { 2 } , 1 + [ x ] \right\}$. Then the value of the integral $\int _ { 0 } ^ { 2 } f ( x ) d x$ is :
(1) $\frac { 5 + 4 \sqrt { 2 } } { 3 }$
(2) $\frac { 8 + 4 \sqrt { 2 } } { 3 }$
(3) $\frac { 1 + 5 \sqrt { 2 } } { 3 }$
(4) $\frac { 4 + 5 \sqrt { 2 } } { 3 }$

Let $[ t ]$ denote the greatest integer $\leq t$. Then $\frac { 2 } { \pi } \int _ { \frac { \pi } { 6 } } ^ { \frac { 5 \pi } { 6 } } ( 8 [ \operatorname { cosec } x ] - 5 [ \cot x ] ) d x$ is equal to $\_\_\_\_$

If $\int _ { - 0.15 } ^ { 0.15 } \left| 100 x ^ { 2 } - 1 \right| d x = \frac { k } { 3000 }$, then $k$ is equal to $\_\_\_\_$ .

Let $[ t ]$ denote the largest integer less than or equal to $t$. If $$\int _ { 0 } ^ { 3 } \left( \left[ x ^ { 2 } \right] + \left[ \frac { x ^ { 2 } } { 2 } \right] \right) \mathrm { d } x = \mathrm { a } + \mathrm { b } \sqrt { 2 } - \sqrt { 3 } - \sqrt { 5 } + \mathrm { c } \sqrt { 6 } - \sqrt { 7 }$$ where $\mathrm { a } , \mathrm { b } , \mathrm { c } \in \mathbf { Z }$, then $\mathrm { a } + \mathrm { b } + \mathrm { c }$ is equal to $\_\_\_\_$

If $24 \int _ { 0 } ^ { \frac { \pi } { 4 } } \left( \sin \left| 4 x - \frac { \pi } { 12 } \right| + [ 2 \sin x ] \right) \mathrm { d } x = 2 \pi + \alpha$, where $[ \cdot ]$ denotes the greatest integer function, then $\alpha$ is equal to $\_\_\_\_$ .

$$f(x) = \begin{cases} 3 - x, & x < 2 \\ 2x - 3, & x \geq 2 \end{cases}$$
What is the value of the integral $\displaystyle\int_{1}^{3} f(x+1)\, dx$?
A) 2
B) 4
C) 6
D) 8
E) 10

$f ( x ) = \begin{cases} 2 x - 4 , & \text{if } 0 \leq x < 1 \\ - 2 , & \text{if } 1 \leq x < 4 \\ x - 6 , & \text{if } 4 \leq x \leq 6 \end{cases}$
Given this, what is the value of the integral $\int _ { 0 } ^ { 6 } f ( x ) d x$?
A) - 11
B) - 10
C) - 9
D) - 8
E) - 7