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 $f ( t ) = \int _ { 0 } ^ { \pi } \frac { 2 x \mathrm {~d} x } { 1 - \cos ^ { 2 } \mathrm { t } \sin ^ { 2 } x } , 0 < \mathrm { t } < \pi$, then the value of $\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \pi ^ { 2 } \mathrm { dt } } { f ( \mathrm { t } ) }$ equals $\_\_\_\_$

Let $f ( x ) = \int _ { 0 } ^ { x } t \left( t ^ { 2 } - 9 t + 20 \right) d t , 1 \leq x \leq 5$. If the range of $f$ is $[ \alpha , \beta ]$, then $4 ( \alpha + \beta )$ equals:
(1) 253
(2) 154
(3) 125
(4) 157

Let for $f ( x ) = 7 \tan ^ { 8 } x + 7 \tan ^ { 6 } x - 3 \tan ^ { 4 } x - 3 \tan ^ { 2 } x , \quad \mathrm { I } _ { 1 } = \int _ { 0 } ^ { \pi / 4 } f ( x ) \mathrm { d } x$ and $\mathrm { I } _ { 2 } = \int _ { 0 } ^ { \pi / 4 } x f ( x ) \mathrm { d } x$. Then $7 \mathrm { I } _ { 1 } + 12 \mathrm { I } _ { 2 }$ is equal to:
(1) 2
(2) 1
(3) $2 \pi$
(4) $\pi$

Let $f : \mathbf { R } \rightarrow \mathbf { R }$ be a twice differentiable function such that $f ( x + y ) = f ( x ) f ( y )$ for all $x , y \in \mathbf { R }$. If $f ^ { \prime } ( 0 ) = 4 \mathrm { a }$ and $f$ satisfies $f ^ { \prime \prime } ( x ) - 3 \mathrm { a } f ^ { \prime } ( x ) - f ( x ) = 0 , \mathrm { a } > 0$, then the area of the region $\mathrm { R } = \{ ( x , y ) \mid 0 \leq y \leq f ( \mathrm { a } x ) , 0 \leq x \leq 2 \}$ is:
(1) $e ^ { 2 } - 1$
(2) $\mathrm { e } ^ { 2 } + 1$
(3) $e ^ { 4 } + 1$
(4) $e ^ { 4 } - 1$

If $I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1}\,dx$, $m, n > 0$, then $I(9, 14) + I(10, 13)$ is
(1) $I(19, 27)$
(2) $I(9, 1)$
(3) $I(1, 13)$
(4) $I(9, 13)$

If $\mathrm { I } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin ^ { \frac { 3 } { 2 } } x } { \sin ^ { \frac { 3 } { 2 } } x + \cos ^ { \frac { 3 } { 2 } } x } \mathrm {~d} x$, then $\int _ { 0 } ^ { 2\mathrm{I} } \frac { x \sin x \cos x } { \sin ^ { 4 } x + \cos ^ { 4 } x } \mathrm {~d} x$ equals :
(1) $\frac { \pi ^ { 2 } } { 12 }$
(2) $\frac { \pi ^ { 2 } } { 4 }$
(3) $\frac { \pi ^ { 2 } } { 16 }$
(4) $\frac { \pi ^ { 2 } } { 8 }$

If $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { 96 x ^ { 2 } \cos ^ { 2 } x } { \left( 1 + e ^ { x } \right) } \mathrm { d } x = \pi \left( \alpha \pi ^ { 2 } + \beta \right) , \alpha , \beta \in \mathbb { Z }$, then $( \alpha + \beta ) ^ { 2 }$ equals
(1) 64
(2) 196
(3) 144
(4) 100

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 $\_\_\_\_$ .

Q74. Let $\int _ { \alpha } ^ { \log _ { e } 4 } \frac { \mathrm {~d} x } { \sqrt { \mathrm { e } ^ { x } - 1 } } = \frac { \pi } { 6 }$. Then $\mathrm { e } ^ { \alpha }$ and $\mathrm { e } ^ { - \alpha }$ are the roots of the equation :
(1) $x ^ { 2 } + 2 x - 8 = 0$
(2) $x ^ { 2 } - 2 x - 8 = 0$
(3) $2 x ^ { 2 } - 5 x + 2 = 0$
(4) $2 x ^ { 2 } - 5 x - 2 = 0$

Q75. Let $f ( x ) = \left\{ \begin{array} { l l } - 2 , & - 2 \leq x \leq 0 \\ x - 2 , & 0 < x \leq 2 \end{array} \right.$ and $h ( x ) = f ( | x | ) + | f ( x ) |$. Then $\int _ { - 2 } ^ { 2 } h ( x ) \mathrm { d } x$ is equal to :
(1) 1
(2) 6
(3) 4
(4) 2

Q75. If the value of the integral $\int _ { - 1 } ^ { 1 } \frac { \cos \alpha x } { 1 + 3 ^ { x } } d x$ is $\frac { 2 } { \pi }$. Then, a value of $\alpha$ is
(1) $\frac { \pi } { 3 }$
(2) $\frac { \pi } { 6 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 2 }$

Q75. The value of $\int _ { - \pi } ^ { \pi } \frac { 2 y ( 1 + \sin y ) } { 1 + \cos ^ { 2 } y } d y$ is :
(1) $2 \pi ^ { 2 }$
(2) $\frac { \pi ^ { 2 } } { 2 }$
(3) $\frac { \pi } { 2 }$
(4) $\pi ^ { 2 }$

Q75. The value of the integral $\int _ { - 1 } ^ { 2 } \log _ { e } \left( x + \sqrt { x ^ { 2 } + 1 } \right) d x$ is
(1) $\sqrt { 5 } - \sqrt { 2 } + \log _ { e } \left( \frac { 7 + 4 \sqrt { 5 } } { 1 + \sqrt { 2 } } \right)$
(2) $\sqrt { 5 } - \sqrt { 2 } + \log _ { e } \left( \frac { 9 + 4 \sqrt { 5 } } { 1 + \sqrt { 2 } } \right)$
(3) $\sqrt { 2 } - \sqrt { 5 } + \log _ { e } \left( \frac { 7 + 4 \sqrt { 5 } } { 1 + \sqrt { 2 } } \right)$
(4) $\sqrt { 2 } - \sqrt { 5 } + \log _ { e } \left( \frac { 9 + 4 \sqrt { 5 } } { 1 + \sqrt { 2 } } \right)$

Q87. If $\int _ { 0 } ^ { \frac { \pi } { 4 } } \frac { \sin ^ { 2 } x } { 1 + \sin x \cos x } \mathrm {~d} x = \frac { 1 } { \mathrm { a } } \log _ { \mathrm { e } } \left( \frac { \mathrm { a } } { 3 } \right) + \frac { \pi } { \mathrm { b } \sqrt { 3 } }$, where $\mathrm { a } , \mathrm { b } \in \mathbf { N }$, then $\mathrm { a } + \mathrm { b }$ is equal to $\_\_\_\_$

Q87. 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 $\_\_\_\_$

The value of $\int_{0}^{\pi/2} |\sin x + \sin 2x + \sin 3x|dx$ is
(A) 17 (B) 16 (C) 15 (D) 14

The value of $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { 12 ( 3 + [ x ] ) } { 3 + [ \sin x ] + [ \cos x ] } d x$ is equal to (A) $3 + 10 \pi$ (B) $11 \pi + 2$ (C) $10 \pi + 2$

$\int _ { \frac { - \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { d x } { [ x ] + 5 }$ is equal to: ([.] denotes greatest integer function)
(A) $\frac { \pi } { 4 } - \frac { 1 } { 10 }$
(B) $\frac { \pi } { 4 } - \frac { 1 } { 20 }$
(C) $\frac { \pi } { 12 } - \frac { 1 } { 10 }$
(D) $\frac { \pi } { 4 } - \frac { 1 } { 5 }$

If $6 \left( \int _ { 1 } ^ { \mathbf { x } } \mathbf { f } ( \mathbf { t } ) \mathbf { d t } \right) = 3 \left( \mathbf { x } \mathbf { f } ( \mathbf { x } ) + \mathbf { x } ^ { 3 } - 4 \right)$, then find the value of $\mathbf { f } ( 2 ) - \mathbf { f } ( 3 )$

Let the area bounded by the curve $\mathrm { f } ( \mathrm { x } ) = \max \{ \sin x , \cos x \}$ and x -axis is\ $A$ where $x \in \left[ 0 , \frac { 3 \pi } { 2 } \right]$. Find $A + A ^ { 2 }$

$\int _ { 0 } ^ { 36 } \mathbf { f } \left( \frac { \mathbf { t x } } { 36 } \right) \mathbf { d t } = \mathbf { 4 \alpha f } ( \mathbf { x } )$
If the curve represented by $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$ is a standard parabola passing through $( 2,1 )$ and $( - 4 , \beta )$ then find

If $\mathbf { f }$ be a real valued function such that $\mathbf { f } \left( \mathbf { x } ^ { \mathbf { 2 } } + \mathbf { 1 } \right) = \mathbf { x } ^ { \mathbf { 4 } } + \mathbf { 5 } \mathbf { x } ^ { \mathbf { 2 } } + \mathbf { 2 }$, then $\int _ { 0 } ^ { 3 } f ( x ) d x$ is equal to (A) 16 (B) $\frac { 31 } { 2 }$ (C) $\frac { 33 } { 2 }$ (D) 14

Find $\int _ { \frac { \pi } { 24 } } ^ { \frac { 5 \pi } { 24 } } \frac { 1 + ( \tan 2 \mathrm { x } ) ^ { 1 / 3 } } { 1 + ( \tan 2 x ) ^ { 1 / 3 } } \mathrm { dx }$\ (A) $\frac { \pi } { 24 }$\ (B) $\frac { \pi } { 12 }$\ (C) $\frac { \pi } { 48 }$\ (D) $\frac { \pi } { 6 }$

The value of $\int_{-\pi/6}^{\pi/6} \left(\frac{\pi + 4x^{11}}{1 - \sin(|x| + \frac{\pi}{6})}\right)dx$ is equal to
(A) $8\pi$ (B) $7\pi$ (C) $5\pi$ (D) $4\pi$