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 $\alpha$ for which $4 \alpha \int _ { - 1 } ^ { 2 } e ^ { - \alpha | x | } d x = 5$, is
(1) $\log _ { e } 2$
(2) $\log _ { e } \left( \frac { 3 } { 2 } \right)$
(3) $\log _ { e } \sqrt { 2 }$
(4) $\log _ { e } \left( \frac { 4 } { 3 } \right)$

If for all real triplets $( a , b , c ) , f ( x ) = a + b x + c x ^ { 2 }$; then $\int _ { 0 } ^ { 1 } f ( x ) d x$ is equal to:
(1) $2 \left\{ 3 f ( 1 ) + 2 f \left( \frac { 1 } { 2 } \right) \right\}$
(2) $\frac { 1 } { 2 } \left\{ f ( 1 ) + 3 f \left( \frac { 1 } { 2 } \right) \right\}$
(3) $\frac { 1 } { 3 } \left\{ f ( 0 ) + f \left( \frac { 1 } { 2 } \right) \right\}$
(4) $\frac { 1 } { 6 } \left\{ f ( 0 ) + f ( 1 ) + 4 f \left( \frac { 1 } { 2 } \right) \right\}$

The integral $\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 3 } } \tan ^ { 3 } x \cdot \sin ^ { 2 } 3 x \left( 2 \sec ^ { 2 } x \cdot \sin ^ { 2 } 3 x + 3 \tan x \cdot \sin 6 x \right) d x$ is equal to:
(1) $\frac { 7 } { 18 }$
(2) $- \frac { 1 } { 9 }$
(3) $- \frac { 1 } { 18 }$
(4) $\frac { 9 } { 2 }$

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

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
(1) $\left[ \frac { 327 } { 360 } , \frac { 329 } { 360 } \right]$
(2) $\left[ \frac { 330 } { 360 } , \frac { 331 } { 360 } \right]$
(3) $\left[ \frac { 331 } { 360 } , \frac { 334 } { 360 } \right]$
(4) $\left[ \frac { 335 } { 360 } , \frac { 336 } { 360 } \right]$

If the real part of the complex number $( 1 - \cos \theta + 2i \sin \theta ) ^ { - 1 }$ is $\frac { 1 } { 5 }$ for $\theta \in ( 0 , \pi )$, then the value of the integral $\int _ { 0 } ^ { \theta } \sin x \mathrm {~d} x$ is equal to:
(1) 1
(2) 2
(3) - 1
(4) 0

The value of the integral $\frac { 48 } { \pi ^ { 4 } } \int _ { 0 } ^ { \pi } \left( \frac { 3 \pi x ^ { 2 } } { 2 } - x ^ { 3 } \right) \frac { \sin x } { 1 + \cos ^ { 2 } x } \, dx$ is equal to $\_\_\_\_$.

The value of the integral $\int _ { 0 } ^ { \frac { \pi } { 2 } } 60 \frac { \sin ( 6 x ) } { \sin x } d x$ is equal to $\_\_\_\_$.

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) $\frac{\pi}{4} - \tan^{-1}(2)$

If $\int_0^{\frac{\pi}{3}} \cos^4 x \, dx = a\pi + b\sqrt{3}$, where $a$ and $b$ are rational numbers, then $9a + 8b$ is equal to:
(1) 2
(2) 1
(3) 3
(4) $\frac{3}{2}$

$\int _ { 0 } ^ { \pi / 4 } \frac { \cos ^ { 2 } x \sin ^ { 2 } x } { \left( \cos ^ { 3 } x + \sin ^ { 3 } x \right) ^ { 2 } } d x$ is equal to
(1) $1 / 6$
(2) $1 / 3$
(3) $1 / 12$
(4) $1 / 9$

If $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 3 + x } + \sqrt { 1 + x } } d x = a + b \sqrt { 2 } + c \sqrt { 3 }$, where $a , b , c$ are rational numbers, then $2 a + 3 b - 4 c$ is equal to:
(1) 4
(2) 10
(3) 7
(4) 8

For $0 < \mathrm { a } < 1$, the value of the integral $\int _ { 0 } ^ { \pi } \frac { d x } { 1 - 2 a \cos x + a ^ { 2 } }$ is :
(1) $\frac { \pi ^ { 2 } } { \pi + a ^ { 2 } }$
(2) $\frac { \pi ^ { 2 } } { \pi - a ^ { 2 } }$
(3) $\frac { \pi } { 1 - a ^ { 2 } }$
(4) $\frac { \pi } { 1 + a ^ { 2 } }$

Let $f: R \rightarrow R$ be a function defined $f(x) = \frac{x}{(1 + x^4)^{1/4}}$ and $g(x) = f(f(f(f(x))))$, then $18\int_0^{\sqrt{2\sqrt{5}}} x^2 g(x)\, dx$
(1) 33
(2) 36
(3) 42
(4) 39

Let $a$ and $b$ be real constants such that the function $f$ defined by $$f(x) = \begin{cases} x^2 + 3x + a, & x \leq 1 \\ bx + 2, & x > 1 \end{cases}$$ be differentiable on $R$. Then, the value of $\int_{-2}^{2} f(x)\, dx$ equals
(1) $\frac{15}{6}$
(2) $\frac{19}{6}$
(3) 21
(4) 17

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

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) $1 / 4$

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

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) $I(9, 13)$

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

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