We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
We set
$$P _ { n } ( x ) = \sum _ { k = 0 } ^ { n } a _ { k } x ^ { k }, \quad ( 1 - y ) ^ { n } = \sum _ { k = 0 } ^ { n } b _ { k } y ^ { k }$$
with for all $k \in \llbracket 0 , n \rrbracket , a _ { k } \in \mathbb { Z }$ and $b _ { k } \in \mathbb { Z }$.
Let $n \in \mathbb { N } ^ { * }$. Justify the existence of
$$I _ { n } = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { ( 1 - y ) ^ { n } P _ { n } ( x ) } { 1 - x y } \mathrm {~d} x \mathrm {~d} y$$
and show that
$$I _ { n } = \sum _ { \substack { r , s = 0 \\ r \neq s } } ^ { n } a _ { r } b _ { s } J _ { r , s } + \sum _ { r = 0 } ^ { n } a _ { r } b _ { r } J _ { r , r }$$

Problem 1: calculation of an integral
For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Deduce the value of $\int _ { 0 } ^ { \infty } \frac { \sin t } { t } \mathrm {~d} t$.

The equation $x^{2} + (b/a)x + (c/a) = 0$ has two real roots $\alpha$ and $\beta$. If $a > 0$, then the area under the curve $f(x) = x^{2} + (b/a)x + (c/a)$ between $\alpha$ and $\beta$ is
(a) $(b^{2} - 4ac)/2a$
(b) $(b^{2} - 4ac)^{3/2}/6a^{3}$
(c) $-(b^{2} - 4ac)^{3/2}/6a^{3}$
(d) $-(b^{2} - 4ac)/2a$

The value(s) of $\int _ { 0 } ^ { 1 } \frac { x ^ { 4 } ( 1 - x ) ^ { 4 } } { 1 + x ^ { 2 } } d x$ is (are)
A) $\frac { 22 } { 7 } - \pi$
B) $\frac { 2 } { 105 }$
C) 0
D) $\frac { 71 } { 15 } - \frac { 3 \pi } { 2 }$

Let $f ( x ) = 7 \tan ^ { 8 } x + 7 \tan ^ { 6 } x - 3 \tan ^ { 4 } x - 3 \tan ^ { 2 } x$ for all $x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$. Then the correct expression(s) is(are)
(A) $\quad \int _ { 0 } ^ { \pi / 4 } x f ( x ) d x = \frac { 1 } { 12 }$
(B) $\quad \int _ { 0 } ^ { \pi / 4 } f ( x ) d x = 0$
(C) $\int _ { 0 } ^ { \pi / 4 } x f ( x ) d x = \frac { 1 } { 6 }$
(D) $\int _ { 0 } ^ { \pi / 4 } f ( x ) d x = 1$

If $\int _ { 1 } ^ { 3 } x ^ { 2 } F ^ { \prime \prime } ( x ) d x = - 12$ and $\int _ { 1 } ^ { 3 } x ^ { 3 } F ^ { \prime \prime } ( x ) d x = 40$, then the correct expression(s) is(are)
(A) $9 f ^ { \prime } ( 3 ) + f ^ { \prime } ( 1 ) - 32 = 0$
(B) $\int _ { 1 } ^ { 3 } f ( x ) d x = 12$
(C) $9 f ^ { \prime } ( 3 ) - f ^ { \prime } ( 1 ) + 32 = 0$
(D) $\int _ { 1 } ^ { 3 } f ( x ) d x = - 12$

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0) = 0$, $f\left(\frac{\pi}{2}\right) = 3$ and $f'(0) = 1$. If $$g(x) = \int_x^{\frac{\pi}{2}} \left[f'(t)\operatorname{cosec} t - \cot t\operatorname{cosec} t\, f(t)\right] dt$$ for $x \in \left(0, \frac{\pi}{2}\right]$, then $\lim_{x \rightarrow 0} g(x) =$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that its derivative $f'$ is continuous and $f(\pi) = -6$.
If $F: [0, \pi] \rightarrow \mathbb{R}$ is defined by $F(x) = \int_{0}^{x} f(t)\, dt$, and if $$\int_{0}^{\pi} \left(f'(x) + F(x)\right) \cos x\, dx = 2$$ then the value of $f(0)$ is $\_\_\_\_$

Let $g_i : [\pi/8, 3\pi/8] \to \mathbb{R}$, $i = 1, 2$, and $f : [\pi/8, 3\pi/8] \to \mathbb{R}$ be functions such that $$g_1(x) = 1, \quad g_2(x) = |4x - \pi|, \quad f(x) = \sin^2 x,$$ for all $x \in [\pi/8, 3\pi/8]$. Define $$S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx, \quad i = 1, 2.$$
The value of $\frac{16 S_1}{\pi}$ is ____.

Let $g_i : [\pi/8, 3\pi/8] \to \mathbb{R}$, $i = 1, 2$, and $f : [\pi/8, 3\pi/8] \to \mathbb{R}$ be functions such that $$g_1(x) = 1, \quad g_2(x) = |4x - \pi|, \quad f(x) = \sin^2 x,$$ for all $x \in [\pi/8, 3\pi/8]$. Define $$S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx, \quad i = 1, 2.$$
The value of $\frac{48 S_2}{\pi^2}$ is ____.

Let $g _ { i } : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R } , i = 1,2$, and $f : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R }$ be functions such that $$g _ { 1 } ( x ) = 1 , g _ { 2 } ( x ) = | 4 x - \pi | \text { and } f ( x ) = \sin ^ { 2 } x , \text { for all } x \in \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right]$$ Define $$S _ { i } = \int _ { \frac { \pi } { 8 } } ^ { \frac { 3 \pi } { 8 } } f ( x ) \cdot g _ { i } ( x ) d x , \quad i = 1,2$$ The value of $\frac { 16 S _ { 1 } } { \pi }$ is $\_\_\_\_$.

Let $g _ { i } : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R } , i = 1,2$, and $f : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R }$ be functions such that $$g _ { 1 } ( x ) = 1 , g _ { 2 } ( x ) = | 4 x - \pi | \text { and } f ( x ) = \sin ^ { 2 } x , \text { for all } x \in \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right]$$ Define $$S _ { i } = \int _ { \frac { \pi } { 8 } } ^ { \frac { 3 \pi } { 8 } } f ( x ) \cdot g _ { i } ( x ) d x , \quad i = 1,2$$ The value of $\frac { 48 S _ { 2 } } { \pi ^ { 2 } }$ is $\_\_\_\_$.

The greatest integer less than or equal to
$$\int _ { 1 } ^ { 2 } \log _ { 2 } \left( x ^ { 3 } + 1 \right) d x + \int _ { 1 } ^ { \log _ { 2 } 9 } \left( 2 ^ { x } - 1 \right) ^ { \frac { 1 } { 3 } } d x$$
is $\_\_\_\_$ .

Let $f : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0,1 ]$ be the function defined by $f ( x ) = \sin ^ { 2 } x$ and let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0 , \infty )$ be the function defined by $g ( x ) = \sqrt { \frac { \pi x } { 2 } - x ^ { 2 } }$. The value of $2 \int _ { 0 } ^ { \frac { \pi } { 2 } } f ( x ) g ( x ) d x - \int _ { 0 } ^ { \frac { \pi } { 2 } } g ( x ) d x$ is $\_\_\_\_$ .

Let $f : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0,1 ]$ be the function defined by $f ( x ) = \sin ^ { 2 } x$ and let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0 , \infty )$ be the function defined by $g ( x ) = \sqrt { \frac { \pi x } { 2 } - x ^ { 2 } }$. The value of $\frac { 16 } { \pi ^ { 3 } } \int _ { 0 } ^ { \frac { \pi } { 2 } } f ( x ) g ( x ) d x$ is $\_\_\_\_$ .

The integral $\int _ { 0 } ^ { \pi } \sqrt { 1 + 4 \sin ^ { 2 } \frac { x } { 2 } - 4 \sin \frac { x } { 2 } } \, d x$ equals
(1) $4 \sqrt { 3 } - 4$
(2) $4 \sqrt { 3 } - 4 - \frac { \pi } { 3 }$
(3) $\pi - 4$
(4) $\frac { 2 \pi } { 3 } - 4 - 4 \sqrt { 3 }$

The integral $\int_{\pi/4}^{3\pi/4} \frac{dx}{1 + \cos x}$ is equal to:
(1) $-1$
(2) $-2$
(3) $2$
(4) $4$

The integral $\int_{\pi/4}^{3\pi/4} \frac{dx}{1+\cos x}$ is equal to: (1) $-1$ (2) $-2$ (3) $2$ (4) $4$

The integral $\int _ { \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 4 } } \frac { d x } { 1 + \cos x }$ is equal to:
(1) $- 1$
(2) $- 2$
(3) 2
(4) 4

The integral $\displaystyle\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x}$ is equal to
(1) $-2$
(2) $2$
(3) $4$
(4) $-1$

The integral $\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 4 } } \frac { 8 \cos 2 x } { ( \tan x + \cot x ) ^ { 3 } } d x$ equals
(1) $\frac { 13 } { 256 }$
(2) $\frac { 15 } { 64 }$
(3) $\frac { 13 } { 32 }$
(4) $\frac { 15 } { 128 }$

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) $\log _ { e } \left( \frac { 4 } { 3 } \right)$