We denote by $f$ the function defined on the interval $[ 0 ; \pi ]$ by

$$f ( x ) = \mathrm { e } ^ { x } \sin ( x )$$

We denote by $\mathscr { C } _ { f }$ the representative curve of $f$ in a coordinate system.

\section*{PART A}
\begin{enumerate}
  \item a. Prove that for every real number $x$ in the interval $[ 0 ; \pi ]$,
$$f ^ { \prime } ( x ) = \mathrm { e } ^ { x } [ \sin ( x ) + \cos ( x ) ]$$
b. Justify that the function $f$ is strictly increasing on the interval $\left[ 0 ; \frac { \pi } { 2 } \right]$
  \item a. Determine an equation of the tangent $T$ to the curve $\mathscr { C } _ { f }$ at the point with abscissa 0.\\
b. Prove that the function $f$ is convex on the interval $\left[ 0 ; \frac { \pi } { 2 } \right]$.\\
c. Deduce that for every real number $x$ in the interval $\left[ 0 ; \frac { \pi } { 2 } \right] , \mathrm { e } ^ { x } \sin ( x ) \geqslant x$.
  \item Justify that the point with abscissa $\frac { \pi } { 2 }$ of the representative curve of the function $f$ is an inflection point.
\end{enumerate}

\section*{PART B}
We denote

$$I = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { x } \sin ( x ) \mathrm { d } x \text { and } \quad J = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { x } \cos ( x ) \mathrm { d } x$$

\begin{enumerate}
  \item By integrating by parts the integral $I$ in two different ways, establish the following two relations:
$$I = 1 + J \quad \text { and } \quad I = \mathrm { e } ^ { \frac { \pi } { 2 } } - J$$
  \item Deduce that $I = \frac { 1 + \mathrm { e } ^ { \frac { \pi } { 2 } } } { 2 }$.
  \item We denote by $g$ the function defined on $\mathbb { R }$ by $g ( x ) = x$.\\
Calculate the exact value of the area of the shaded region situated between the curves $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ and the lines with equations $x = 0$ and $x = \frac { \pi } { 2 }$.
\end{enumerate}