Consider the function $f$ defined on the interval $[ 0 ; 1 ]$ by:

$$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { 1 - x } }$$

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

\section*{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

$$u _ { n } = \int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x$$

\begin{enumerate}
  \item 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 $f _ { 0 }$.
  \item Let $n$ be a natural number, interpret graphically $u _ { n }$ and specify the value of $u _ { 0 }$.
  \item What conjecture can be made regarding the direction of variation of the sequence $\left( u _ { n } \right)$ ?
\end{enumerate}

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