We consider the function $f$ defined on $\mathbb { R }$ by:

$$f ( x ) = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 }$$

Let A be a point on $\mathscr { C }$ with positive abscissa $a$. The rotation around the x-axis applied to the part of $\mathscr { C }$ bounded by points I and A generates a surface modeling the flute container, taking 1 cm as the unit.

The real number $a$ being strictly positive, we admit that the volume $V ( a )$ of this solid in $\mathrm { cm } ^ { 3 }$ is given by the formula:

$$V ( a ) = \pi \int _ { 0 } ^ { a } ( f ( x ) ) ^ { 2 } \mathrm {~d} x$$

\begin{enumerate}
  \item Verify, for all real numbers $x \geqslant 0$, the equality:
$$( f ( x ) ) ^ { 2 } = 4 \left( \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 } + \frac { - \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } } \right) .$$
  \item Determine a primitive on $\mathbb { R }$ of each of the functions:
$$g : x \longmapsto \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 } \quad \text { and } \quad h : x \longmapsto \frac { - \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } }$$
  \item Deduce that for all real $a > 0$:
$$V ( a ) = 4 \pi \left[ \ln \left( \frac { \mathrm { e } ^ { a } + 1 } { 2 } \right) + \frac { 1 } { \mathrm { e } ^ { a } + 1 } - \frac { 1 } { 2 } \right] .$$
  \item Using a calculator, determine an approximate value of $a$ to 0.1, knowing that a flute must contain $12.5 \mathrm { cL }$, that is $125 \mathrm {~cm} ^ { 3 }$. No justification is required.
\end{enumerate}