A continuous function $f ( x )$ satisfies
$$f ( x ) = e ^ { x ^ { 2 } } + \int _ { 0 } ^ { 1 } t f ( t ) d t$$
What is the value of $\int _ { 0 } ^ { 1 } x f ( x ) d x$? [3 points]
(1) $e - 2$
(2) $\frac { e - 1 } { 2 }$
(3) $\frac { e } { 2 }$
(4) $e - 1$
(5) $\frac { e + 1 } { 2 }$

26. Let $f ( x ) = \int ^ { x } { } _ { 1 } \sqrt { } \left( 2 - t ^ { 2 } \right)$ The real roots of the equation $x ^ { 2 } - f ^ { \prime } ( x ) = 0$ are
(A) $\quad \pm 1$
(B) $\pm 1 / \sqrt { } 2$
(C) $\quad \pm 1 / 2$
(D) 0 and 1

7. If $\int \sin x ^ { 1 } t ^ { 2 } f ( t ) d t = 1 - \sin x \forall x \hat { I } [ 0 , \Pi / 2 ]$ then $f ( 1 / \sqrt { } 3 )$ is:
(a) 3
(b) $\sqrt { } 3$
(c) $1 / 3$
(d) none of these

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) $\sec x + x \tan x - \frac { 1 } { 2 }$

Let $g : ( 0 , \infty ) \rightarrow R$ be a differentiable function such that $\int \left( \frac { x \cos x - \sin x } { e ^ { x } + 1 } + \frac { g(x) ( e ^ { x } + 1 ) - x e ^ { x } } { ( e ^ { x } + 1 ) ^ { 2 } } \right) \mathrm { d } x = \frac { x g(x) } { e ^ { x } + 1 } + C$, for all $x > 0$, where $C$ is an arbitrary constant. Then
(1) $g$ is decreasing in $\left( 0 , \frac { \pi } { 4 } \right)$
(2) $g - g ^ { \prime }$ is increasing in $\left( 0 , \frac { \pi } { 2 } \right)$
(3) $g ^ { \prime }$ is increasing in $\left( 0 , \frac { \pi } { 4 } \right)$
(4) $g + g ^ { \prime }$ is increasing in $\left( 0 , \frac { \pi } { 2 } \right)$

If $\int \frac { 1 } { \mathrm { a } ^ { 2 } \sin ^ { 2 } x + \mathrm { b } ^ { 2 } \cos ^ { 2 } x } \mathrm {~d} x = \frac { 1 } { 12 } \tan ^ { - 1 } ( 3 \tan x ) +$ constant, then the maximum value of $\mathrm { a } \sin x + \mathrm { b } \cos x$, is:
(1) $\sqrt { 40 }$
(2) $\sqrt { 41 }$
(3) $\sqrt { 39 }$
(4) $\sqrt { 42 }$

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) $\frac { \pi } { 6 } \sqrt { \frac { e } { 2 } }$

Q88. Let $r _ { k } = \frac { \int _ { 0 } ^ { 1 } \left( 1 - x ^ { 7 } \right) ^ { k } d x } { \int _ { 0 } ^ { 1 } \left( 1 - x ^ { 7 } \right) ^ { k + 1 } d x } , k \in \mathbb { N }$. Then the value of $\sum _ { k = 1 } ^ { 10 } \frac { 1 } { 7 \left( r _ { k } - 1 \right) }$ is equal to $\_\_\_\_$

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

A sequence $u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots$ is defined as follows:
$$\begin{aligned} & u _ { 0 } = 1 \\ & u _ { n } = \int _ { 0 } ^ { 1 } 4 x u _ { n - 1 } d x \quad \text { for } n \geqslant 1 \end{aligned}$$
What is the value of $u _ { 1000 }$ ?
A $2 ^ { 1000 }$
B $4 ^ { 1000 }$
C $\frac { 4 } { 1000 ! }$
D $\frac { 4 } { 1001 ! }$
$\mathbf { E } \quad \frac { 2 ^ { 1000 } } { 1000 ! }$
F $\frac { 4 ^ { 1000 } } { 1000 ! }$
G $\frac { 2 ^ { 1000 } } { 1001 ! }$
$\mathbf { H } \frac { 4 ^ { 1000 } } { 1001 ! }$