\section*{Exercise 2}
We consider the function $f$ defined and differentiable on the set $\mathbb { R }$ of real numbers by

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

We denote by $\mathscr { C }$ its representative curve in an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ).

\section*{Part A}
\begin{enumerate}
  \item Let $g$ be the function defined and differentiable on the set $\mathbb { R }$ by
$$g ( x ) = 1 - x + \mathrm { e } ^ { x }$$
Draw up, by justifying, the table giving the variations of the function $g$ on $\mathbb { R }$ (the limits of $g$ at the boundaries of its domain are not required).\\
Deduce the sign of $g ( x )$.
  \item Determine the limit of $f$ at $- \infty$ then the limit of $f$ at $+ \infty$.
  \item We call $f ^ { \prime }$ the derivative of the function $f$ on $\mathbb { R }$. Prove that, for all real $x$,
$$f ^ { \prime } ( x ) = \mathrm { e } ^ { - x } g ( x )$$
  \item Deduce the variation table of the function $f$ on $\mathbb { R }$.
  \item Prove that the equation $f ( x ) = 0$ admits a unique real solution $\alpha$ on $\mathbb { R }$.\\
Prove that $- 1 < \alpha < 0$.
  \item a. Prove that the line T with equation $y = 2 x + 1$ is tangent to the curve $\mathscr { C }$ at the point with abscissa 0.\\
b. Study the relative position of the curve $\mathscr { C }$ and the line T.
\end{enumerate}

\section*{Part B}
\begin{enumerate}
  \item Let H be the function defined and differentiable on $\mathbb { R }$ by
$$\mathrm { H } ( x ) = ( - x - 1 ) \mathrm { e } ^ { - x }.$$
Prove that H is a primitive on $\mathbb { R }$ of the function $h$ defined by $h ( x ) = x \mathrm { e } ^ { - x }$.
  \item We denote by $\mathscr { D }$ the domain bounded by the curve $\mathscr { C }$, the line T and the lines with equations $x = 1$ and $x = 3$.\\
Calculate, in square units, the area of the domain $\mathscr { D }$.
\end{enumerate}