Let the function $f : [ 1 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f ( t ) = \left\{ \begin{array} { c c } ( - 1 ) ^ { n + 1 } 2 , & \text { if } t = 2 n - 1 , n \in \mathbb { N } , \\ \frac { ( 2 n + 1 - t ) } { 2 } f ( 2 n - 1 ) + \frac { ( t - ( 2 n - 1 ) ) } { 2 } f ( 2 n + 1 ) , & \text { if } 2 n - 1 < t < 2 n + 1 , n \in \mathbb { N } . \end{array} \right.$$ Define $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t , x \in ( 1 , \infty )$. Let $\alpha$ denote the number of solutions of the equation $g ( x ) = 0$ in the interval $( 1,8 ]$ and $\beta = \lim _ { x \rightarrow 1 + } \frac { g ( x ) } { x - 1 }$. Then the value of $\alpha + \beta$ is equal to $\_\_\_\_$ .

Let $f : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0,1 ]$ be the function defined by $f ( x ) = \sin ^ { 2 } x$ and let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0 , \infty )$ be the function defined by $g ( x ) = \sqrt { \frac { \pi x } { 2 } - x ^ { 2 } }$. The value of $2 \int _ { 0 } ^ { \frac { \pi } { 2 } } f ( x ) g ( x ) d x - \int _ { 0 } ^ { \frac { \pi } { 2 } } g ( x ) d x$ is $\_\_\_\_$ .

Let $f : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0,1 ]$ be the function defined by $f ( x ) = \sin ^ { 2 } x$ and let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0 , \infty )$ be the function defined by $g ( x ) = \sqrt { \frac { \pi x } { 2 } - x ^ { 2 } }$. The value of $\frac { 16 } { \pi ^ { 3 } } \int _ { 0 } ^ { \frac { \pi } { 2 } } f ( x ) g ( x ) d x$ is $\_\_\_\_$ .

Let $F ( x ) = f ( x ) + f \left( \frac { 1 } { x } \right)$, where $f ( x ) = \int _ { 1 } ^ { x } \frac { \log t } { 1 + t } d t$. Then $F ( e )$ equals
(1) $\frac { 1 } { 2 }$
(2) 0
(3) 1
(4) 2

If the integral $\displaystyle\int_{0}^{10} \frac{\lfloor x \rfloor e^{x}}{e^{\lfloor x \rfloor}} dx = \alpha(e-1)$, then $\alpha$ is equal to (where $\lfloor x \rfloor$ denotes the greatest integer function)
(1) $\frac{1}{e-1}$
(2) $\frac{10}{e-1}$
(3) $\frac{e}{e-1}$
(4) $\frac{e^{10}-1}{e-1}$

Statement-I: The value of the integral $\int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}}$ is equal to $\frac{\pi}{6}$. Statement-II: $\int_a^b f(x)\, dx = \int_a^b f(a + b - x)\, dx$.
(1) Statement-I is true; Statement-II is false.
(2) Statement-I is false; Statement-II is true.
(3) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
(4) Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.

The integral $\int_{\pi/4}^{3\pi/4} \frac{dx}{1 + \cos x}$ is equal to:
(1) $-1$
(2) $-2$
(3) $2$
(4) $4$

The integral $\int _ { 2 } ^ { 4 } \frac { \log x ^ { 2 } } { \log x ^ { 2 } + \log ( 6 - x ) ^ { 2 } } d x$ is equal to
(1) 6
(2) 2
(3) 4
(4) 1

The integral $\int_{\pi/4}^{3\pi/4} \frac{dx}{1+\cos x}$ is equal to: (1) $-1$ (2) $-2$ (3) $2$ (4) $4$

The value of the integral $\int _ { 4 } ^ { 10 } \frac { \left[ x ^ { 2 } \right] } { \left[ x ^ { 2 } - 28 x + 196 \right] + \left[ x ^ { 2 } \right] } d x$, where $[ x ]$ denotes the greatest integer less than or equal to $x$, is
(1) $\frac { 1 } { 3 }$
(2) 6
(3) 7
(4) 3

The integral $\int_0^{\pi/4} \frac{\sin x + \cos x}{9 + 16\sin 2x} dx$ is equal to:
(1) $\frac{1}{20} \log 3$
(2) $\log 3$
(3) $\frac{1}{20} \log 9$
(4) $\frac{1}{10} \log 3$

If $f : \mathbb { R } \to \mathbb { R }$ is a differentiable function and $f ( 2 ) = 6$, then $\lim _ { x \to 2 } \int _ { 6 } ^ { f ( x ) } \frac { 2 t \, d t } { ( x - 2 ) }$ is:
(1) $2 f ^ { \prime } ( 2 )$
(2) $12 f ^ { \prime } ( 2 )$
(3) $0$
(4) $24 f ^ { \prime } ( 2 )$

The integral $\displaystyle\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x}$ is equal to
(1) $-2$
(2) $2$
(3) $4$
(4) $-1$

The integral $\int _ { \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 4 } } \frac { d x } { 1 + \cos x }$ is equal to:
(1) $- 1$
(2) $- 2$
(3) 2
(4) 4

The values of $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { \sin ^ { 2 } x } { 1 + 2 ^ { x } } d x$ is
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 8 }$
(3) $\frac { \pi } { 2 }$
(4) $4 \pi$

The value of the integral $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \sin ^ { 4 } x \left( 1 + \ln \left( \frac { 2 + \sin x } { 2 - \sin x } \right) \right) d x$ is
(1) $\frac { 3 } { 4 }$
(2) $\frac { 3 } { 8 } \pi$
(3) 0
(4) $\frac { 3 } { 16 } \pi$

The value of the integral
$$\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \sin ^ { 4 } x \left( 1 + \log \left( \frac { 2 + \sin x } { 2 - \sin x } \right) \right) d x$$
is
(1) $\frac { 3 } { 16 } \pi$
(2) 0
(3) $\frac { 3 } { 8 } \pi$
(4) $\frac { 3 } { 4 }$

Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x) - f(y)| \leq 2|x-y|^{3/2}$, for all $x,y \in R$. If $f(0) = 1$ then $\int_0^1 f^2(x)\,dx$ is equal to
(1) 0
(2) 1
(3) 2
(4) $\frac{1}{2}$

The integral $\int _ { 1 } ^ { e } \left\{ \left( \frac { x } { e } \right) ^ { 2 x } - \left( \frac { e } { x } \right) ^ { x } \right\} \log _ { e } x \, d x$ is equal to
(1) $\frac { 3 } { 2 } - e - \frac { 1 } { 2 e ^ { 2 } }$
(2) $\frac { 1 } { 2 } - e - \frac { 1 } { e ^ { 2 } }$
(3) $- \frac { 1 } { 2 } + \frac { 1 } { e } - \frac { 1 } { 2 e ^ { 2 } }$
(4) $\frac { 3 } { 2 } - \frac { 1 } { e } - \frac { 1 } { 2 e ^ { 2 } }$

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

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

If $\int_0^{\pi/3} \frac{\tan\theta}{\sqrt{2k\sec\theta}}\,d\theta = 1 - \frac{1}{\sqrt{2}},\,(k > 0)$, then the value of $k$ is
(1) $\frac{1}{2}$
(2) 1
(3) 2
(4) 4

Let $f ( x ) = \int _ { 0 } ^ { x } g ( t ) \, dt$, where $g$ is a non-zero even function. If $f ( x + 5 ) = g ( x )$, then $\int _ { 0 } ^ { x } f ( t ) \, dt$ equals
(1) $\int _ { 5 } ^ { x + 5 } g ( t ) \, dt$
(2) $\int _ { x + 5 } ^ { 5 } g ( t ) \, dt$
(3) $5 \int _ { x + 5 } ^ { 5 } g ( t ) \, dt$
(4) $2 \int _ { 5 } ^ { x + 5 } g ( t ) \, dt$

On the $x$-axis and at a distance $x$ from the origin, the gravitational field due to a mass distribution is given by $\frac { A x } { \left( x ^ { 2 } + a ^ { 2 } \right) ^ { 3 / 2 } }$ in the $x$-direction. The magnitude of the gravitational potential on the $x$-axis at a distance $x$, taking its value to be zero at infinity is:
(1) $\frac { A } { \left( x ^ { 2 } + a ^ { 2 } \right) ^ { 1 / 2 } }$
(2) $\frac { A } { \left( x ^ { 2 } + a ^ { 2 } \right) ^ { 3 / 2 } }$
(3) $A \left( x ^ { 2 } + a ^ { 2 } \right) ^ { 1 / 2 }$
(4) $A \left( x ^ { 2 } + a ^ { 2 } \right) ^ { 3 / 2 }$

If $f(a + b + 1 - x) = f(x)$, for all $x$, where $a$ and $b$ are fixed positive real numbers, then $\frac { 1 } { a + b } \int _ { a } ^ { b } x (f(x) + f(x + 1)) d x$ is equal to
(1) $\int _ { a - 1 } ^ { b - 1 } f(x + 1) d x$
(2) $\int _ { a - 1 } ^ { b - 1 } f(x) d x$
(3) $\int _ { a + 1 } ^ { b + 1 } f(x) d x$
(4) $\int _ { a + 1 } ^ { b + 1 } f(x + 1) d x$