The integral $\int \frac { 1 } { \sqrt [ 4 ] { ( x - 1 ) ^ { 3 } ( x + 2 ) ^ { 5 } } } \mathrm {~d} x$ is equal to : (where $C$ is a constant of integration)
(1) $\frac { 3 } { 4 } \left( \frac { x + 2 } { x - 1 } \right) ^ { \frac { 5 } { 4 } } + C$
(2) $\frac { 4 } { 3 } \left( \frac { x - 1 } { x + 2 } \right) ^ { \frac { 1 } { 4 } } + C$
(3) $\frac { 4 } { 3 } \left( \frac { x - 1 } { x + 2 } \right) ^ { \frac { 5 } { 4 } } + \mathrm { C }$
(4) $\frac { 3 } { 4 } \left( \frac { x + 2 } { x - 1 } \right) ^ { \frac { 1 } { 4 } } + \mathrm { C }$

$\int \frac { 2 e ^ { x } + 3 e ^ { - x } } { 4 e ^ { x } + 7 e ^ { - x } } d x = \frac { 1 } { 14 } \left( u x + v \log _ { e } \left( 4 e ^ { x } + 7 e ^ { - x } \right) \right) + C$, where $C$ is a constant of integration, then $u + v$ is equal to

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

If $\int \frac { 1 } { x } \sqrt { \frac { 1 - x } { 1 + x } } d x = g ( x ) + c , g ( 1 ) = 0$, then $g \left( \frac { 1 } { 2 } \right)$ is equal to
(1) $\log _ { e } \left( \frac { \sqrt { 3 } - 1 } { \sqrt { 3 } + 1 } \right) + \frac { \pi } { 3 }$
(2) $\log _ { \mathrm { e } } \left( \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 } \right) + \frac { \pi } { 3 }$
(3) $\log _ { \mathrm { e } } \left( \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 } \right) - \frac { \pi } { 3 }$
(4) $\frac { 1 } { 3 } \log _ { e } \left( \frac { \sqrt { 3 } - 1 } { \sqrt { 3 } + 1 } \right) - \frac { \pi } { 6 }$

Let $f$ be a differentiable function in $\left( 0 , \frac { \pi } { 2 } \right)$. If $\int _ { \cos x } ^ { 1 } t ^ { 2 } f ( t ) d t = \sin ^ { 3 } x + \cos x$, then $\frac { 1 } { \sqrt { 3 } } f ^ { \prime } \left( \frac { 1 } { \sqrt { 3 } } \right)$ is equal to
(1) $6 - 9 \sqrt { 2 }$
(2) $6 + \frac { 9 } { \sqrt { 2 } }$
(3) $6 - \frac { 9 } { \sqrt { 2 } }$
(4) $3 + \sqrt { 2 }$

The integral $\int \frac{ \left(1 - \frac { 1 } { \sqrt { 3 } }\right) \cos x - \sin x } { 1 + \frac { 2 } { \sqrt { 3 } } \sin 2 x } d x$ is equal to
(1) $\frac { 1 } { 2 } \log _ { e } \left| \frac { \tan \left( \frac { x } { 2 } + \frac { \pi } { 12} \right) } { \tan\left( \frac { x } { 2 } + \frac { \pi } { 6 } \right) } \right| + C$
(2) $\log _ { e } \left| \frac { \tan \left( \frac { x } { 2 } + \frac { \pi } { 6 } \right) } { \tan\left( \frac { x } { 2 } + \frac { \pi } { 3 } \right) } \right| + C$
(3) $\frac { 1 } { 2 } \log _ { e } \left| \frac { \tan \left( \frac { x } { 2 } + \frac { \pi } { 6 } \right) } { \tan\left( \frac { x } { 2 } + \frac { \pi } { 3 } \right) } \right| + C$
(4) $\frac { 1 } { 2 } \log _ { e } \left| \frac { \tan \left( \frac { x } { 2 } - \frac { \pi } { 12 } \right) } { \tan \left( \frac { x } { 2 } - \frac { \pi } { 6 } \right) } \right| + C$

The slope of the tangent to a curve $C : y = y(x)$ at any point $(x,y)$ on it is $\frac { 2e ^ { 2x } - 6e ^ { -x } + 9 } { 2 + 9e ^ { -2x } }$. If $C$ passes through the points $\left(0, \frac{1}{2} + \frac{\pi}{2\sqrt{2}}\right)$ and $\left(\alpha, \frac{1}{2} e ^ { 2\alpha }\right)$ then $e ^ { \alpha }$ is equal to
(1) $\frac { 3 + \sqrt { 2 } } { 3 - \sqrt { 2 } }$
(2) $\frac { 3 } { \sqrt { 2 } } \cdot \frac { 3 + \sqrt { 2 } } { 3 - \sqrt { 2 } }$
(3) $\frac { 1 } { \sqrt { 2 } } \cdot \frac { \sqrt { 2 } + 1 } { \sqrt { 2 } - 1 }$
(4) $\frac { \sqrt { 2 } + 1 } { \sqrt { 2 } - 1 }$

If $I(x) = \int e^{\sin^2 x} \cos x (\sin 2x - \sin x)\, dx$ and $I(0) = 1$, then $I\!\left(\frac{\pi}{3}\right)$ is equal to
(1) $-\frac{1}{2}e^{\frac{3}{4}}$
(2) $\frac{1}{2}e^{\frac{3}{4}}$
(3) $-e^{\frac{3}{4}}$
(4) $e^{\frac{3}{4}}$

Let $[ x ]$ denote the greatest integer function and $f ( x ) = \max \{ 1 + x + [ x ] , 2 + x , x + 2 [ x ] \} , 0 \leq x \leq 2$, where $m$ is the number of points in $( 0,2 )$ where $f$ is not continuous and $n$ be the number of points in $( 0,2 )$, where $f$ is not differentiable. Then $( m + n ) ^ { 2 } + 2$ is equal to
(1) 2
(2) 11
(3) 6
(4) 3

If $\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 5 + 2 x - 2 x ^ { 2 } \right) \left( 1 + e ^ { ( 2 - 4 x ) } \right) } d x = \frac { 1 } { \alpha } \log _ { e } \left( \frac { \alpha + 1 } { \beta } \right) , \alpha , \beta > 0$, then $\alpha ^ { 4 } - \beta ^ { 4 }$ is equal to
(1) 19
(2) $- 21$
(3) 0
(4) 21

Let $\alpha \in (0,1)$ and $\beta = \log_e(1-\alpha)$. Let $P_n(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \ldots + \frac{x^n}{n}$, $x \in (0,1)$. Then the integral $\int_0^{\alpha} \frac{t^{50}}{1-t}\,dt$ is equal to
(1) $\beta - P_{50}(\alpha)$
(2) $-\beta + P_{50}(\alpha)$
(3) $P_{50}(\alpha) - \beta$
(4) $\beta + P_{50}(\alpha)$

The value of $\int_{\pi/3}^{\pi/2} \frac{2 + 3\sin x}{\sin x(1 + \cos x)}\,dx$ is equal to
(1) $\frac{7}{2} - \sqrt{3} - \log_e\sqrt{3}$
(2) $-2 + 3\sqrt{3} + \log_e\sqrt{3}$
(3) $\frac{10}{3} - \sqrt{3} + \log_e\sqrt{3}$
(4) $\frac{10}{3} - \sqrt{3} - \log_e\sqrt{3}$

The integral $16 \int _ { 1 } ^ { 2 } \frac { d x } { x ^ { 3 } \left( x ^ { 2 } + 2 \right) ^ { 2 } }$ is equal to
(1) $\frac { 11 } { 6 } + \log _ { e } 4$
(2) $\frac { 11 } { 12 } + \log _ { e } 4$
(3) $\frac { 11 } { 12 } - \log _ { e } 4$
(4) $\frac { 11 } { 6 } - \log _ { e } 4$

If $\int \sqrt{\sec 2x - 1}\, dx = \alpha \log_{e}\left|\cos 2x + \beta + \sqrt{\cos 2x\left(1 + \cos \frac{1}{\beta}x\right)}\right| + \text{constant}$, then $\beta - \alpha$ is equal to $\_\_\_\_$.

Let $I ( x ) = \int \sqrt { \frac { x + 7 } { x } } d x$ and $I ( 9 ) = 12 + 7 \log _ { e } 7$. If $I ( 1 ) = \alpha + 7 \log _ { e } ( 1 + 2 \sqrt { 2 } )$, then $\alpha ^ { 4 }$ is equal to $\_\_\_\_$ .

Let $f ( x ) = \int \frac { 2 x } { \left( x ^ { 2 } + 1 \right) \left( x ^ { 2 } + 3 \right) } d x$. If $f ( 3 ) = \frac { 1 } { 2 } \left( \log _ { e } 5 - \log _ { e } 6 \right)$, then $f ( 4 )$ is equal to
(1) $\frac { 1 } { 2 } \left( \log _ { e } 17 - \log _ { e } 19 \right)$
(2) $\log _ { \mathrm { e } } 17 - \log _ { \mathrm { e } } 18$
(3) $\frac { 1 } { 2 } \left( \log _ { e } 19 - \log _ { e } 17 \right)$
(4) $\log _ { e } 19 - \log _ { e } 20$

If $\int _ { \frac { 1 } { 3 } } ^ { 3 } \left| \log _ { e } x \right| dx = \frac { m } { n } \log _ { e } \left( \frac { n ^ { 2 } } { e } \right)$, where m and n are coprime natural numbers, then $m ^ { 2 } + n ^ { 2 } - 5$ is equal to $\_\_\_\_$.

Let $I ( x ) = \int \frac { x + 1 } { x \left( 1 + x e ^ { x } \right) ^ { 2 } } d x , x > 0$. If $\lim _ { x \rightarrow \infty } I ( x ) = 0$ then $I ( 1 )$ is equal to
(1) $\frac { e + 2 } { e + 1 } - \log _ { e } ( e + 1 )$
(2) $\frac { e + 1 } { e + 2 } + \log _ { e } ( e + 1 )$
(3) $\frac { e + 1 } { e + 2 } - \log _ { e } ( e + 1 )$
(4) $\frac { e + 2 } { e + 1 } + \log _ { e } ( e + 1 )$

If $\int_0^1 (x^{21} + x^{14} + x^7)(2x^{14} + 3x^7 + 6)^{1/7}\, dx = \frac{1}{l} \cdot 11^{m/n}$ where $l, m, n \in \mathbb{N}$, $m$ and $n$ are co-prime, then $l + m + n$ is equal to $\_\_\_\_$.

The integral $\int \frac { x ^ { 8 } - x ^ { 2 } d x } { x ^ { 12 } + 3 x ^ { 6 } + 1 \tan ^ { - 1 } x ^ { 3 } + \frac { 1 } { x ^ { 3 } } }$ is equal to :
(1) $\log \tan ^ { - 1 } x ^ { 3 } + { \frac { 1 } { x ^ { 3 } } } ^ { \frac { 1 } { 3 } } + C$
(2) $\log _ { e } \tan ^ { - 1 } x ^ { 3 } + { \frac { 1 } { x ^ { 3 } } } ^ { \frac { 1 } { 2 } } + C$
(3) $\log _ { e } \tan ^ { - 1 } x ^ { 3 } + \frac { 1 } { x ^ { 3 } } + C$
(4) $\log _ { e } \tan ^ { - 1 } x ^ { 3 } + { \frac { 1 } { x ^ { 3 } } } ^ { 3 } + C$

If $\int \frac { \sin ^ { \frac { 3 } { 2 } } x + \cos ^ { \frac { 3 } { 2 } } x } { \sqrt { \sin ^ { 3 } x \cos ^ { 3 } x \sin ( x - \theta ) } } d x = A \sqrt { \cos \theta \tan x - \sin \theta } + B \sqrt { \cos \theta - \sin \theta \cot x } + C$, where $C$ is the integration constant, then $AB$ is equal to
(1) $4 \operatorname { cosec } ( 2 \theta )$
(2) $4 \sec \theta$
(3) $2 \sec \theta$
(4) $8 \operatorname { cosec } ( 2 \theta )$

For $x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$, if $y ( x ) = \int \frac { \operatorname { cosec } x + \sin x } { \operatorname { cosec } x \sec x + \tan x \sin ^ { 2 } x } d x$ and $\lim _ { x \rightarrow \left( \frac { \pi } { 2 } \right) ^ { - } } y ( x ) = 0$ then $y \left( \frac { \pi } { 4 } \right)$ is equal to
(1) $\tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right)$
(2) $\frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right)$
(3) $- \frac { 1 } { \sqrt { 2 } } \tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right)$
(4) $\frac { 1 } { \sqrt { 2 } } \tan ^ { - 1 } \left( - \frac { 1 } { 2 } \right)$

Let $y = f(x)$ be a thrice differentiable function on $(-5, 5)$. Let the tangents to the curve $y = f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\frac{\pi}{6}$ and $\frac{\pi}{4}$, respectively with positive $x$-axis. If $27\int_1^3 \left(f'(t)\right)^2 + 1\right) f''(t)\, dt = \alpha + \beta\sqrt{3}$ where $\alpha, \beta$ are integers, then the value of $\alpha + \beta$ equals
(1) $-14$
(2) 26
(3) $-16$
(4) 36

If $( a , b )$ be the orthocentre of the triangle whose vertices are $( 1,2 ) , ( 2,3 )$ and $( 3,1 )$, and $I _ { 1 } = \int _ { \mathrm { a } } ^ { \mathrm { b } } \mathrm { x } \sin \left( 4 \mathrm { x } - \mathrm { x } ^ { 2 } \right) \mathrm { dx } , \mathrm { I } _ { 2 } = \int _ { \mathrm { a } } ^ { \mathrm { b } } \sin \left( 4 \mathrm { x } - \mathrm { x } ^ { 2 } \right) \mathrm { dx }$, then $36 \frac { I _ { 1 } } { I _ { 2 } }$ is equal to:
(1) 72
(2) 88
(3) 80
(4) 66

If $\int \frac { 1 } { \sqrt [ 5 ] { ( x - 1 ) ^ { 4 } ( x + 3 ) ^ { 6 } } } \mathrm {~d} x = \mathrm { A } \left( \frac { \alpha x - 1 } { \beta x + 3 } \right) ^ { B } + \mathrm { C }$, where C is the constant of integration, then the value of $\alpha + \beta + 20 \mathrm { AB }$ is $\_\_\_\_$