Consider the function $f$ defined on the interval $[ 0 ; 1 ]$ by:
$$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { 1 - x } }$$

Part A

1. Study the direction of variation of the function $f$ on the interval $[ 0 ; 1 ]$.
2. Prove that for all real $x$ in the interval $[ 0 ; 1 ] , f ( x ) = \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + \mathrm { e } }$ (recall that $\mathrm { e } = \mathrm { e } ^ { 1 }$ ).
3. Show then that $\int _ { 0 } ^ { 1 } f ( x ) \mathrm { d } x = \ln ( 2 ) + 1 - \ln ( 1 + \mathrm { e } )$.

Part B

Let $n$ be a natural number. Consider the functions $f _ { n }$ defined on $[ 0 ; 1 ]$ by:
$$f _ { n } ( x ) = \frac { 1 } { 1 + n \mathrm { e } ^ { 1 - x } }$$
We denote $\mathscr { C } _ { n }$ the representative curve of the function $f _ { n }$ in the plane with an orthonormal coordinate system. Consider the sequence with general term
$$u _ { n } = \int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x$$

1. The representative curves of the functions $f _ { n }$ for $n$ varying from 1 to 5 are drawn in the appendix. Complete the graph by drawing the curve $\mathscr { C } _ { 0 }$ representative of the function $f _ { 0 }$.
2. Let $n$ be a natural number, interpret graphically $u _ { n }$ and specify the value of $u _ { 0 }$.
3. What conjecture can be made regarding the direction of variation of the sequence $\left( u _ { n } \right)$ ?

Prove this conjecture.
4. Does the sequence ( $u _ { n }$ ) have a limit?

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The two endomorphisms $T$ and $M$ of $E$ are defined by $$\forall P \in \mathbb{R}_{2m}[X], \quad T(P) = P' \text{ and } M(P) = P^*$$ where $P^*(X) = P(-X)$. The map $S$ is defined by $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$.
Show that $$\forall (P,Q) \in E^2, \quad S(P,Q) = P(1)Q(1) - P(-1)Q(-1)$$

Let $I_n = \int \tan^n x\, dx$ $(n > 1)$. If $I_4 + I_6 = a\tan^5 x + bx^5 + c$, then the ordered pair $(a, b)$ is equal to
(1) $\left(-\dfrac{1}{5}, 1\right)$
(2) $\left(\dfrac{1}{5}, 0\right)$
(3) $\left(\dfrac{1}{5}, -1\right)$
(4) $\left(-\dfrac{1}{5}, 0\right)$

If $\int \frac { d \theta } { \cos ^ { 2 } \theta ( \tan 2 \theta + \sec 2 \theta ) } = \lambda \tan \theta + 2 \log _ { e } | f ( \theta ) | + C$ where $C$ is a constant of integration, then the ordered pair $( \lambda , f ( \theta ) )$ is equal to:
(1) $( 1,1 - \tan \theta )$
(2) $( - 1,1 - \tan \theta )$
(3) $( - 1,1 + \tan \theta )$
(4) $( 1,1 + \tan \theta )$