For a real number $a$, when $\int _ { - a } ^ { a } \left( 3 x ^ { 2 } + 2 x \right) d x = \frac { 1 } { 4 }$, find the value of $50 a$. [3 points]

What is the value of $\int _ { 0 } ^ { 1 } 3 \sqrt { x } \, d x$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

If $\int _ { 0 } ^ { 1 } ( 2 x + a ) d x = 4$, what is the value of the constant $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For the function $f ( x ) = \frac { 1 } { x }$, what is the value of $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } f \left( 1 + \frac { 2 k } { n } \right) \frac { 2 } { n }$? [3 points]
(1) $\ln 6$
(2) $\ln 5$
(3) $2 \ln 2$
(4) $\ln 3$
(5) $\ln 2$

The function $f ( x )$ satisfies $f ( x + 3 ) = f ( x )$ for all real numbers $x$, and $$f ( x ) = \begin{cases} x & ( 0 \leq x < 1 ) \\ 1 & ( 1 \leq x < 2 ) \\ - x + 3 & ( 2 \leq x < 3 ) \end{cases}$$ If $\int _ { - a } ^ { a } f ( x ) d x = 13$, what is the value of the constant $a$? [4 points]
(1) 10
(2) 12
(3) 14
(4) 16
(5) 18

The derivative $f ^ { \prime } ( x )$ of a polynomial function $f ( x )$ is $f ^ { \prime } ( x ) = 6 x ^ { 2 } + 4$. If the graph of $y = f ( x )$ passes through the point $( 0,6 )$, find the value of $f ( 1 )$. [4 points]

A quadratic function $f ( x )$ satisfies $f ( 0 ) = 0$ and the following conditions. (가) $\int _ { 0 } ^ { 2 } | f ( x ) | d x = - \int _ { 0 } ^ { 2 } f ( x ) d x = 4$ (나) $\int _ { 2 } ^ { 3 } | f ( x ) | d x = \int _ { 2 } ^ { 3 } f ( x ) d x$ Find the value of $f ( 5 )$. [4 points]

What is the value of $\int _ { 0 } ^ { \frac { \pi } { 2 } } 2 \sin x \, d x$? [2 points]
(1) 0
(2) $\frac { 1 } { 2 }$
(3) 1
(4) $\frac { 3 } { 2 }$
(5) 2

What is the value of $\int _ { 0 } ^ { 2 } \left( 6 x ^ { 2 } - x \right) d x$? [3 points]
(1) 15
(2) 14
(3) 13
(4) 12
(5) 11

What is the value of $\int _ { 1 } ^ { e } \ln \frac { x } { e } \, d x$? [3 points]
(1) $\frac { 1 } { e } - 1$
(2) $2 - e$
(3) $\frac { 1 } { e } - 2$
(4) $1 - e$
(5) $\frac { 1 } { 2 } - e$

Find the positive value of $a$ that satisfies $\int _ { 0 } ^ { a } \left( 3 x ^ { 2 } - 4 \right) d x = 0$. [3 points]
(1) 2
(2) $\frac { 9 } { 4 }$
(3) $\frac { 5 } { 2 }$
(4) $\frac { 11 } { 4 }$
(5) 3

When the function $f ( x )$ is $$f ( x ) = \int _ { 0 } ^ { x } \frac { 1 } { 1 + e ^ { - t } } d t$$ what is the value of the real number $a$ that satisfies $( f \circ f ) ( a ) = \ln 5$? [4 points]
(1) $\ln 11$
(2) $\ln 13$
(3) $\ln 15$
(4) $\ln 17$
(5) $\ln 19$

For the quadratic function $f ( x ) = \frac { 3 x - x ^ { 2 } } { 2 }$, a function $g ( x )$ defined on the interval $[ 0 , \infty )$ satisfies the following conditions. (가) When $0 \leq x < 1$, $g ( x ) = f ( x )$. (나) When $n \leq x < n + 1$, $$\begin{aligned} & g ( x ) = \frac { 1 } { 2 ^ { n } } \{ f ( x - n ) - ( x - n ) \} + x \\ & \text{(Here, } n \text{ is a natural number.)} \end{aligned}$$ For some natural number $k ( k \geq 6 )$, the function $h ( x )$ is defined as $$h ( x ) = \begin{cases} g ( x ) & ( 0 \leq x < 5 \text{ or } x \geq k ) \\ 2 x - g ( x ) & ( 5 \leq x < k ) \end{cases}$$ When the sequence $\left\{ a _ { n } \right\}$ is defined by $a _ { n } = \int _ { 0 } ^ { n } h ( x ) d x$ and $\lim _ { n \rightarrow \infty } \left( 2 a _ { n } - n ^ { 2 } \right) = \frac { 241 } { 768 }$, find the value of $k$. [4 points]

A polynomial function $f ( x )$ satisfies for all real numbers $x$: $$\int _ { 1 } ^ { x } \left\{ \frac { d } { d t } f ( t ) \right\} d t = x ^ { 3 } + a x ^ { 2 } - 2$$ What is the value of $f ^ { \prime } ( a )$? (Here, $a$ is a constant.) [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Find the value of $\int _ { 1 } ^ { 4 } ( x + | x - 3 | ) d x$. [3 points]

For the function $f ( x ) = 4 x ^ { 3 } + x$, what is the value of $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 1 } { n } f \left( \frac { 2 k } { n } \right)$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

A polynomial function $f ( x )$ satisfies the following conditions. (가) For all real numbers $x$, $$\int _ { 1 } ^ { x } f ( t ) d t = \frac { x - 1 } { 2 } \{ f ( x ) + f ( 1 ) \}$$ (나) $\int _ { 0 } ^ { 2 } f ( x ) d x = 5 \int _ { - 1 } ^ { 1 } x f ( x ) d x$ When $f ( 0 ) = 1$, find the value of $f ( 4 )$. [4 points]

What is the area of the region enclosed by the curve $y = e ^ { 2 x }$, the $x$-axis, and the two lines $x = \ln \frac { 1 } { 2 }$ and $x = \ln 2$? [3 points]
(1) $\frac { 5 } { 3 }$
(2) $\frac { 15 } { 8 }$
(3) $\frac { 15 } { 7 }$
(4) $\frac { 5 } { 2 }$
(5) 3

What is the value of $\lim _ { n \rightarrow \infty } \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \sqrt { \frac { 3 n } { 3 n + k } }$? [3 points]
(1) $4 \sqrt { 3 } - 6$
(2) $\sqrt { 3 } - 1$
(3) $5 \sqrt { 3 } - 8$
(4) $2 \sqrt { 3 } - 3$
(5) $3 \sqrt { 3 } - 5$

For the function $f ( x ) = \pi \sin 2 \pi x$, a function $g ( x )$ with domain being the set of all real numbers and range being the set $\{ 0,1 \}$, and a natural number $n$ satisfy the following conditions. What is the value of $n$? [4 points]
The function $h ( x ) = f ( n x ) g ( x )$ is continuous on the set of all real numbers and $$\int _ { - 1 } ^ { 1 } h ( x ) d x = 2 , \quad \int _ { - 1 } ^ { 1 } x h ( x ) d x = - \frac { 1 } { 32 }$$
(1) 8
(2) 10
(3) 12
(4) 14
(5) 16

For a function $f ( x )$, if $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 4 x + 5$ and $f ( 0 ) = 4$, find the value of $f ( 1 )$. [3 points]

The position $x ( t )$ of a point P moving on a number line at time $t$ is given by $$x ( t ) = t ( t - 1 ) ( a t + b ) \quad ( a \neq 0 )$$ for two constants $a , b$. The velocity $v ( t )$ of point P at time $t$ satisfies $\int _ { 0 } ^ { 1 } | v ( t ) | d t = 2$. Which of the following statements in the given options are correct? [4 points]
Given statements: ᄀ. $\int _ { 0 } ^ { 1 } v ( t ) d t = 0$ ㄴ. There exists $t _ { 1 }$ in the open interval $( 0,1 )$ such that $\left| x \left( t _ { 1 } \right) \right| > 1$. ㄷ. If $| x ( t ) | < 1$ for all $t$ with $0 \leq t \leq 1$, then there exists $t _ { 2 }$ in the open interval $( 0,1 )$ such that $x \left( t _ { 2 } \right) = 0$.
(1) ᄀ
(2) ᄀ, ㄴ
(3) ᄀ, ㄷ
(4) ㄴ, ㄷ
(5) ᄀ, ㄴ, ㄷ

For a function $f ( x )$ with $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 2 x$ and $f ( 0 ) = 2$, find the value of $f ( 1 )$. [3 points]

A function $f ( x )$ differentiable on the entire set of real numbers satisfies the following conditions.
(a) On the closed interval $[ 0,1 ]$, $f ( x ) = x$.
(b) For some constants $a , b$, on the interval $[ 0 , \infty )$, $f ( x + 1 ) - x f ( x ) = a x + b$. Find the value of $60 \times \int _ { 1 } ^ { 2 } f ( x ) d x$. [4 points]

A function $f ( x )$ that is continuous on the set of all real numbers satisfies the following condition. When $n - 1 \leq x < n$, $| f ( x ) | = | 6 ( x - n + 1 ) ( x - n ) |$. (Here, $n$ is a natural number.)
For the function $$g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t - \int _ { x } ^ { 4 } f ( t ) d t$$ defined on the open interval $(0, 4)$, when $g ( x )$ has a minimum value of 0 at $x = 2$, what is the value of $\int _ { \frac { 1 } { 2 } } ^ { 4 } f ( x ) d x$? [4 points]
(1) $- \frac { 3 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 3 } { 2 }$
(5) $\frac { 5 } { 2 }$