For the function $f(x) = x + 1$, $$\int _ { -1 } ^ { 1 } \{ f(x) \} ^ { 2 } dx = k \left( \int _ { -1 } ^ { 1 } f(x) dx \right) ^ { 2 }$$ what is the value of the constant $k$? [3 points]
(1) $\frac{1}{6}$
(2) $\frac{1}{3}$
(3) $\frac{1}{2}$
(4) $\frac{2}{3}$
(5) $\frac{5}{6}$

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]

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

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

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

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

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

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

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

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

As shown in the figure, a solid figure has as its base the region enclosed by the curve $y = \sqrt{\frac{x+1}{x(x + \ln x)}}$, the $x$-axis, and the two lines $x = 1$ and $x = e$. When the cross-section of this solid figure cut by a plane perpendicular to the $x$-axis is a square, what is the volume of this solid figure? [3 points]
(1) $\ln(e+1)$
(2) $\ln(e+2)$
(3) $\ln(e+3)$
(4) $\ln(2e+1)$
(5) $\ln(2e+2)$

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

We consider a rectangle $]a,b[ \times ]c,d[$ of the plane $\mathbb{R}^{2}$, with $a < b$ and $c < d$. Calculate the real number $V(]a,b[ \times ]c,d[)$. What does it represent? (One may use functions of the type $$(x,y) \mapsto f(x,y) = \phi(x)\varphi(y)$$ where $\phi$ and $\varphi$ are well-chosen continuous and piecewise affine functions).

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$
Show that the map $(P, Q) \mapsto \langle P, Q \rangle$ is an inner product on $\mathbb{R}[X]$.

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: $$\left\{\begin{array}{l} L_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad L_n = \frac{1}{P_n^{(n)}(1)} P_n^{(n)} \end{array}\right.$$
II.D.1) For all $n \in \mathbb{N}$, we set $I_n = \int_0^1 P_n(u) \, du$.
Calculate, for all $n \in \mathbb{N}$, the value of $I_n$.
II.D.2) Deduce for all $n \in \mathbb{N}$ the relation: $\langle L_n, L_n \rangle = \frac{1}{2n+1}$.

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$ The family $\left(K_n\right)_{n \in \mathbb{N}}$ is the unique family of polynomials such that for all $n \in \mathbb{N}$, the degree of $K_n$ equals $n$ with strictly positive leading coefficient, and for all $N \in \mathbb{N}$, $\left(K_n\right)_{0 \leqslant n \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for $\langle \cdot, \cdot \rangle$.
Calculate $K_0$, $K_1$ and $K_2$.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ and we denote by $\|\cdot\|$ the associated norm.
Let $n \in \mathbb{N}$. Show that there exists a unique polynomial $\Pi_n \in \mathbb{R}_n[X]$ such that $$\left\|\Pi_n - f\right\| = \min_{Q \in \mathbb{R}_n[X]} \|Q - f\|$$

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ For each $n \in \mathbb{N}$, $\Pi_n$ denotes the unique polynomial in $\mathbb{R}_n[X]$ minimizing $\|Q - f\|$ over $\mathbb{R}_n[X]$.
Determine explicitly $\Pi_2$ when $f$ is the function defined for all $t \in [0,1]$ by $f(t) = \frac{1}{1+t^2}$.

For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$ We define, for all $n \in \mathbb{N}^*$, the polynomial $S_n$ by: $S_n = a_0^{(n)} + a_1^{(n)} X + \cdots + a_{n-1}^{(n)} X^{n-1}$, where $\left(a_p^{(n)}\right)_{0 \leqslant p \leqslant n-1}$ is the unique solution of the system in IV.A.2.
Show that $$\forall Q = \alpha_0 + \alpha_1 X + \cdots + \alpha_{n-1} X^{n-1} \in \mathbb{R}_{n-1}[X], \quad \langle S_n, Q \rangle = \sum_{p=0}^{n-1} \alpha_p$$

Deduce that, for all $x \in ] - 1,1 [$, we have $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta = 0$.
Deduce the value of $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ in the case $| x | > 1$.

Deduce that $\int _ { 0 } ^ { \pi } \ln ( \sin \theta ) \mathrm { d } \theta = - \pi \ln 2$.

Deduce that $\int _ { 0 } ^ { \pi } \ln ( 2 - 2 \cos \theta ) \mathrm { d } \theta = \int _ { 0 } ^ { \pi } \ln ( 2 + 2 \cos \theta ) \mathrm { d } \theta = 0$.

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Deduce that $$f ( x ) = \begin{cases} 2 \pi \ln ( | x | ) & \text { if } | x | > 1 \\ 0 & \text { if } | x | < 1 \end{cases}$$
One will first determine coefficients $A$ and $B$ as functions of $x$ such that $\frac { ( x + 1 ) T + ( x - 1 ) } { \left( ( x + 1 ) ^ { 2 } T + ( x - 1 ) ^ { 2 } \right) ( T + 1 ) } = \frac { A } { ( x + 1 ) ^ { 2 } T + ( x - 1 ) ^ { 2 } } + \frac { B } { T + 1 }$ for all $T \in \mathbb { R }$ such that these fractions are defined.