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

A cubic function $f ( x )$ with positive leading coefficient satisfies the following conditions. (가) The function $f ( x )$ has a local maximum at $x = 0$ and a local minimum at $x = k$. (Here, $k$ is a constant.) (나) For all real numbers $t$ greater than 1, $\int _ { 0 } ^ { t } \left| f ^ { \prime } ( x ) \right| d x = f ( t ) + f ( 0 )$ Which of the following statements in the given options are correct? [4 points] Options ᄀ. $\int _ { 0 } ^ { k } f ^ { \prime } ( x ) d x < 0$ ㄴ. $0 < k \leq 1$ ㄷ. The local minimum value of the function $f ( x )$ is 0.
(1) ᄀ
(2) ㄷ
(3) ᄀ, ㄴ
(4) ㄴ, ㄷ
(5) ᄀ, ㄴ, ㄷ

A continuous increasing function $f ( x )$ on the closed interval $[ 0,1 ]$ satisfies $$\int _ { 0 } ^ { 1 } f ( x ) d x = 2 , \quad \int _ { 0 } ^ { 1 } | f ( x ) | d x = 2 \sqrt { 2 }$$ When the function $F ( x )$ is defined as $$F ( x ) = \int _ { 0 } ^ { x } | f ( t ) | d t \quad ( 0 \leq x \leq 1 )$$ what is the value of $\int _ { 0 } ^ { 1 } f ( x ) F ( x ) d x$? [4 points]
(1) $4 - \sqrt { 2 }$
(2) $2 + \sqrt { 2 }$
(3) $5 - \sqrt { 2 }$
(4) $1 + 2 \sqrt { 2 }$
(5) $2 + 2 \sqrt { 2 }$

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

For a real number $t$, define the function $f ( x )$ as $$f ( x ) = \left\{ \begin{array} { c c } 1 - | x - t | & ( | x - t | \leq 1 ) \\ 0 & ( | x - t | > 1 ) \end{array} \right.$$ For a certain odd number $k$, the function $$g ( t ) = \int _ { k } ^ { k + 8 } f ( x ) \cos ( \pi x ) d x$$ satisfies the following condition.
When all $\alpha$ for which the function $g ( t )$ has a local minimum at $t = \alpha$ and $g ( \alpha ) < 0$ are listed in increasing order as $\alpha _ { 1 } , \alpha _ { 2 } , \cdots , \alpha _ { m }$ (where $m$ is a natural number), we have $\sum _ { i = 1 } ^ { m } \alpha _ { i } = 45$. Find the value of $k - \pi ^ { 2 } \sum _ { i = 1 } ^ { m } g \left( \alpha _ { i } \right)$. [4 points]

A continuous function $f ( x )$ defined on $x > 0$ satisfies $$2 f ( x ) + \frac { 1 } { x ^ { 2 } } f \left( \frac { 1 } { x } \right) = \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } }$$ for all positive $x$. What is the value of $\int _ { \frac { 1 } { 2 } } ^ { 2 } f ( x ) d x$? [4 points]
(1) $\frac { \ln 2 } { 3 } + \frac { 1 } { 2 }$
(2) $\frac { 2 \ln 2 } { 3 } + \frac { 1 } { 2 }$
(3) $\frac { \ln 2 } { 3 } + 1$
(4) $\frac { 2 \ln 2 } { 3 } + 1$
(5) $\frac { 2 \ln 2 } { 3 } + \frac { 3 } { 2 }$

An increasing continuous function $f ( x )$ on the set of all real numbers satisfies the following conditions. (가) For all real numbers $x$, $f ( x ) = f ( x - 3 ) + 4$. (나) $\int _ { 0 } ^ { 6 } f ( x ) d x = 0$ What is the area enclosed by the graph of $y = f ( x )$, the $x$-axis, and the two lines $x = 6$ and $x = 9$? [4 points]
(1) 9
(2) 12
(3) 15
(4) 18
(5) 21

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]

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

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

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 cubic function $f(x)$ satisfies $$xf(x) - f(x) = 3x^4 - 3x$$ for all real numbers $x$. Find the value of $\int_{-2}^{2} f(x)\,dx$. [3 points]
(1) 12
(2) 16
(3) 20
(4) 24
(5) 28

A function $f(x)$ is continuous on the set of all real numbers, $f(x) \geq 0$ for all real numbers $x$, and $f(x) = -4xe^{4x^2}$ for $x < 0$. For all positive numbers $t$, the equation $f(x) = t$ has exactly 2 distinct real roots. Let $g(t)$ denote the smaller root and $h(t)$ denote the larger root of this equation. The two functions $g(t)$ and $h(t)$ satisfy $$2g(t) + h(t) = k \quad (k \text{ is a constant})$$ for all positive numbers $t$. If $\int_0^7 f(x)\,dx = e^4 - 1$, find the value of $\frac{f(9)}{f(8)}$. [4 points]
(1) $\frac{3}{2}e^5$
(2) $\frac{4}{3}e^7$
(3) $\frac{5}{4}e^9$
(4) $\frac{6}{5}e^{11}$
(5) $\frac{7}{6}e^{13}$

For the function $f(x) = 3x^{2} - 16x - 20$, $$\int_{-2}^{a} f(x)\, dx = \int_{-2}^{0} f(x)\, dx$$ When this condition is satisfied, what is the value of the positive number $a$? [4 points]
(1) 16
(2) 14
(3) 12
(4) 10
(5) 8

For the function $f ( x ) = 4 x ^ { 3 } - 2 x$, let $F ( x )$ be an antiderivative with $F ( 0 ) = 4$. Find the value of $F ( 2 )$. [3 points]

11. $\int _ { 0 } ^ { 2 } ( x - 1 ) d x = $ $\_\_\_\_$.

We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$.
Show that, for every pair $(f, g)$ of $E \times E$, the function: $$x \mapsto f(x) g(x) \frac{1}{\sqrt{1 - x^2}}$$ is integrable on the interval $]-1,1[$.

We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$.
The previous question shows that the following application $\varphi$ is well defined: $$\varphi : \left\{ \begin{aligned} E \times E & \rightarrow \mathbb{R} \\ (f, g) & \mapsto \int_{-1}^{1} \frac{1}{\sqrt{1 - x^2}} f(x) g(x)\, dx \end{aligned} \right.$$
Show that $\varphi$ defines an inner product on $E$.

We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$, for all $n \in \mathbb{N}$, $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$, and the space $E$ is equipped with the inner product $(\cdot|\cdot)$ defined by $\varphi(f,g) = \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x) g(x)\, dx$.
a) Show that there exists a sequence of polynomial functions $(p_n)_{n \in \mathbb{N}}$ such that, for all $n \in \mathbb{N}$, $p_n$ has degree $n$ and leading coefficient 1, and that, for all $n \in \mathbb{N}^*$, $p_n$ is orthogonal to all elements of $E_{n-1}$.
b) Show that there exists a unique family $(Q_n)_{n \in \mathbb{N}}$ of polynomial functions satisfying the following conditions:

• [i)] the family $(Q_n)_{n \in \mathbb{N}}$ is orthogonal for the inner product $(\cdot|\cdot)$;
• [ii)] for all $n \in \mathbb{N}$, $Q_n$ has degree $n$ and leading coefficient 1.

We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$, for all $n \in \mathbb{N}$, $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$. For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ and $T_0(x) = 1$. The space $E$ is equipped with the inner product $(\cdot|\cdot)$ defined by $(f|g) = \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x) g(x)\, dx$.
Calculate $(T_m | T_n)$ for all $(m, n) \in \mathbb{N} \times \mathbb{N}$. What can we deduce from this?

Show that the function $t \rightarrow e^{-t} t^{x-1}$ is integrable on $]0, +\infty[$ if, and only if, $x > 0$.

The Euler Gamma function is defined, for all real $x > 0$, by: $$\Gamma(x) = \int_{0}^{+\infty} e^{-t} t^{x-1} dt$$ Justify that the function $\Gamma$ is of class $\mathcal{C}^{1}$ and strictly positive on $]0, +\infty[$.

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ We define the sequence of polynomials $\left(L_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} L_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad L_n = \frac{1}{P_n^{(n)}(1)} P_n^{(n)} \end{array}\right.$$
Let $n \in \mathbb{N}^*$. Show that, for all $Q \in \mathbb{R}_{n-1}[X]$, $\langle Q, L_n \rangle = 0$.
Hint: you may integrate by parts.