4. We define the number $I = \int _ { 0 } ^ { 1 } f _ { 1 } ( x ) \mathrm { d } x$.

Show that $I = \ln \left( \frac { 1 + \mathrm { e } } { 2 } \right)$. Give a graphical interpretation of $I$.

\section*{Part B}
In this part, we choose $k = - 1$ and we wish to sketch the curve $\mathscr { C } _ { - 1 }$ representing the function $f _ { - 1 }$.\\
For all real $x$, we call $P$ the point on $\mathscr { C } _ { 1 }$ with abscissa $x$ and $M$ the point on $\mathscr { C } _ { - 1 }$ with abscissa $x$. We denote by $K$ the midpoint of segment [ $M P$ ].

\begin{enumerate}
  \item Show that, for all real $x , f _ { 1 } ( x ) + f _ { - 1 } ( x ) = 1$.
  \item Deduce that point $K$ belongs to the line with equation $y = \frac { 1 } { 2 }$.
  \item Sketch the curve $\mathscr { C } _ { - 1 }$ on the APPENDIX, to be returned with your answer sheet.
  \item Deduce the area, in square units, of the region bounded by the curves $\mathscr { C } _ { 1 } , \mathscr { C } _ { - 1 }$, the $y$-axis and the line with equation $x = 1$.
\end{enumerate}

\section*{Part C}
In this part, we do not privilege any particular value of the parameter $k$.\\
For each of the following statements, say whether it is true or false and justify your answer.

\begin{enumerate}
  \item Whatever the value of the real number $k$, the graph of the function $f _ { k }$ is strictly between the lines with equations $y = 0$ and $y = 1$.
  \item Whatever the value of the real $k$, the function $f _ { k }$ is strictly increasing.
  \item For all real
$u _ { n }$ & 4502 & 13378 & 39878 & 119122 & 356342 & 1066978 & 3196838 & 9582322 & 28730582 \\
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\end{tabular}
\end{center}

b. What conjecture can be made concerning the monotonicity of the sequence $\left( u _ { n } \right)$ ?\\