If $f \left( \frac { x - 4 } { x + 2 } \right) = 2 x + 1 , ( x \in R = \{ 1 , - 2 \} )$, then $\int f ( x ) d x$ is equal to (where $C$ is a constant of integration)
(1) $12 \log _ { e } | 1 - x | - 3 x + c$
(2) $- 12 \log _ { e } | 1 - x | - 3 x + c$
(3) $- 12 \log _ { e } | 1 - x | + 3 x + c$
(4) $12 \log _ { e } | 1 - x | + 3 x + c$

If $\int e ^ { \sec x } \left( \sec x \tan x f ( x ) + \left( \sec x \tan x + \sec ^ { 2 } x \right) \right) d x = e ^ { \sec x } f ( x ) + C$, then a possible choice of $f ( x )$ is:
(1) $\sec x - \tan x - \frac { 1 } { 2 }$
(2) $\sec x + \tan x + \frac { 1 } { 2 }$
(3) $x \sec x + \tan x + \frac { 1 } { 2 }$
(4) $\sec x + x \tan x - \frac { 1 } { 2 }$

$\int \frac{\sin\frac{5x}{2}}{\sin\frac{x}{2}} dx$ is equal to
(1) $x + 2\sin x + \sin 2x + c$
(2) $2x + \sin x + \sin 2x + c$
(3) $x + 2\sin x + 2\sin 2x + c$
(4) $2x + \sin x + 2\sin 2x + c$

If $\int \left( e ^ { 2 x } + 2 e ^ { x } - e ^ { - x } - 1 \right) e ^ { \left( e ^ { x } + e ^ { - x } \right) } d x = g ( x ) e ^ { \left( e ^ { x } + e ^ { - x } \right) } + c$, where $c$ is a constant of integration, then $g ( 0 )$ is
(1) $e$
(2) $e ^ { 2 }$
(3) 1
(4) 2

If $f ^ { \prime } ( x ) = \tan ^ { - 1 } ( \sec x + \tan x ) , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$ and $f ( 0 ) = 0$, then $f ( 1 )$ is equal to:
(1) $\frac { \pi + 1 } { 4 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { \pi - 1 } { 4 }$
(4) $\frac { \pi + 2 } { 4 }$

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

Let $g ( t ) = \int _ { - \pi / 2 } ^ { \pi / 2 } \left( \cos \frac { \pi } { 4 } t + f ( x ) \right) d x$, where $f ( x ) = \log _ { e } \left( x + \sqrt { x ^ { 2 } + 1 } \right) , x \in R$. Then which one of the following is correct?
(1) $g ( 1 ) = g ( 0 )$
(2) $\sqrt { 2 } g ( 1 ) = g ( 0 )$
(3) $g ( 1 ) = \sqrt { 2 } g ( 0 )$
(4) $g ( 1 ) + g ( 0 ) = 0$

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

For $I(x) = \int \frac{\sec^2 x - 2022}{\sin^{2022} x} dx$, if $I\left(\frac{\pi}{4}\right) = 2^{1011}$, then
(1) $3^{1010} I\left(\frac{\pi}{3}\right) - I\left(\frac{\pi}{6}\right) = 0$
(2) $3^{1010} I\left(\frac{\pi}{6}\right) - I\left(\frac{\pi}{3}\right) = 0$
(3) $3^{1011} I\left(\frac{\pi}{3}\right) - I\left(\frac{\pi}{6}\right) = 0$
(4) $3^{1011} I\left(\frac{\pi}{6}\right) - I\left(\frac{\pi}{3}\right) = 0$

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) $\frac { 1 } { 3 } \log _ { e } \left( \frac { \sqrt { 3 } - 1 } { \sqrt { 3 } + 1 } \right) - \frac { \pi } { 6 }$

The value of $\int_0^1 \frac{d}{dx}\left[\tan^{-1}\left(\frac{1}{1-x+x^2}\right)\right]dx$ is
(1) $\frac{\pi}{4}$
(2) $\tan^{-1}(2)$
(3) $\frac{\pi}{2} - \tan^{-1}(2)$
(4) $\frac{\pi}{4} - \tan^{-1}(2)$

For $\alpha , \beta , \gamma , \delta \in \mathbb { N }$, if $\int \left( \frac { x^2 e^x + e^{2x} } { x } \log _ { e } x \right) dx = \frac { 1 } { \alpha } \frac { x^{\beta} e^x } { 1 } - \frac { 1 } { \gamma } \frac { e ^ { \delta x } } { x } + C$, where $e = \sum _ { n = 0 } ^ { \infty } \frac { 1 } { n ! }$ and $C$ is constant of integration, then $\alpha + 2\beta + 3\gamma - 4\delta$ is equal to
(1) 1
(2) 4
(3) $-4$
(4) $-8$

Let $f : \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right] \rightarrow R$ be a differentiable function such that $f ( 0 ) = \frac { 1 } { 2 }$, If $\lim _ { x \rightarrow 0 } \frac { x \int _ { 0 } ^ { x } f ( t ) d t } { e ^ { x ^ { 2 } } - 1 } = \alpha$, then $8 \alpha ^ { 2 }$ is equal to :
(1) 16
(2) 2
(3) 1
(4) 4

Let $f ( x ) = \int _ { 0 } ^ { x } \left( t + \sin \left( 1 - e ^ { t } \right) \right) d t , x \in \mathbb { R }$. Then, $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 3 } }$ is equal to
(1) $- \frac { 1 } { 6 }$
(2) $\frac { 2 } { 3 }$
(3) $- \frac { 2 } { 3 }$
(4) $\frac { 1 } { 6 }$

The integral $\int _ { 1 / 4 } ^ { 3 / 4 } \cos \left( 2 \cot ^ { - 1 } \sqrt { \frac { 1 - x } { 1 + x } } \right) d x$ is equal to
(1) $1 / 2$
(2) $- 1 / 2$
(3) $- 1 / 4$
(4) $1 / 4$

Let $I ( x ) = \int \frac { 6 } { \sin ^ { 2 } x ( 1 - \cot x ) ^ { 2 } } d x$. If $I ( 0 ) = 3$, then $I \left( \frac { \pi } { 12 } \right)$ is equal to
(1) $2 \sqrt { 3 }$
(2) $\sqrt { 3 }$
(3) $3 \sqrt { 3 }$
(4) $6 \sqrt { 3 }$

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) $\sqrt { 2 } - \sqrt { 5 } + \log _ { e } \left( \frac { 9 + 4 \sqrt { 5 } } { 1 + \sqrt { 2 } } \right)$

If $\int \operatorname { cosec } ^ { 5 } x \, d x = \alpha \cot x \operatorname { cosec } x \left( \operatorname { cosec } ^ { 2 } x + \frac { 3 } { 2 } \right) + \beta \log _ { e } \left| \tan \frac { x } { 2 } \right| + C$ where $\alpha , \beta \in \mathbb { R }$ and C is the constant of integration, then the value of $8 ( \alpha + \beta )$ equals $\_\_\_\_$

If $\int \mathrm { e } ^ { x } \left( \frac { x \sin ^ { - 1 } x } { \sqrt { 1 - x ^ { 2 } } } + \frac { \sin ^ { - 1 } x } { \left( 1 - x ^ { 2 } \right) ^ { 3 / 2 } } + \frac { x } { 1 - x ^ { 2 } } \right) \mathrm { d } x = \mathrm { g } ( x ) + \mathrm { C }$, where C is the constant of integration, then $g \left( \frac { 1 } { 2 } \right)$ equals :
(1) $\frac { \pi } { 4 } \sqrt { \frac { e } { 3 } }$
(2) $\frac { \pi } { 6 } \sqrt { \frac { e } { 3 } }$
(3) $\frac { \pi } { 4 } \sqrt { \frac { e } { 2 } }$
(4) $\frac { \pi } { 6 } \sqrt { \frac { e } { 2 } }$

Let $f$ be a real valued continuous function defined on the positive real axis such that $g ( x ) = \int _ { 0 } ^ { x } \mathrm { t } f ( \mathrm { t } ) \mathrm { dt }$. If $\mathrm { g } \left( x ^ { 3 } \right) = x ^ { 6 } + x ^ { 7 }$, then value of $\sum _ { r = 1 } ^ { 15 } f \left( \mathrm { r } ^ { 3 } \right)$ is :
(1) 270
(2) 340
(3) 320
(4) 310

The integral $80\int_0^{\frac{\pi}{4}} \left(\frac{\sin\theta + \cos\theta}{9 + 16\sin 2\theta}\right)d\theta$ is equal to:
(1) $3\log_e 4$
(2) $4\log_e 3$
(3) $6\log_e 4$
(4) $2\log_e 3$

Q68. Let $f ( x ) = \int _ { 0 } ^ { x } \left( t + \sin \left( 1 - e ^ { \prime } \right) \right) d t , x \in \mathbb { R }$. Then, $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 3 } }$ is equal to
(1) $- \frac { 1 } { 6 }$
(2) $\frac { 2 } { 3 }$
(3) $- \frac { 2 } { 3 }$
(4) $\frac { 1 } { 6 }$

Q73. Let $I ( x ) = \int \frac { 6 } { \sin ^ { 2 } x ( 1 - \cot x ) ^ { 2 } } d x$. If $I ( 0 ) = 3$, then $I \left( \frac { \pi } { 12 } \right)$ is equal to
(1) $2 \sqrt { 3 }$
(2) $\sqrt { 3 }$
(3) $3 \sqrt { 3 }$
(4) $6 \sqrt { 3 }$

Q74. Let $\beta ( \mathrm { m } , \mathrm { n } ) = \int _ { 0 } ^ { 1 } x ^ { \mathrm { m } - 1 } ( 1 - x ) ^ { \mathrm { n } - 1 } \mathrm {~d} x , \mathrm {~m} , \mathrm { n } > 0$. If $\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 10 } \right) ^ { 20 } \mathrm {~d} x = \mathrm { a } \times \beta ( \mathrm { b } , \mathrm { c } )$, then $100 ( \mathrm { a } + \mathrm { b } + \mathrm { c } )$ equals
(1) 1021
(2) 2120
(3) 2012
(4) 1120

Q74. The value of $k \in \mathrm {~N}$ for which the integral $I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 - x ^ { k } \right) ^ { n } d x , n \in \mathbb { N }$, satisfies $147 I _ { 20 } = 148 I _ { 21 }$ is
(1) 14
(2) 8
(3) 10
(4) 7