For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

\begin{enumerate}
  \item We consider the function $f$ defined on $]0; +\infty[$ by: $f(x) = x\ln(x)$.

  Statement 1:
  $$\int_1^{\mathrm{e}} f(x)\,\mathrm{d}x = \frac{\mathrm{e}^2 + 1}{4}$$

  \item Let $n$ and $k$ be two non-zero natural integers such that $k \leqslant n$.

  Statement 2:
  $$n \times \binom{n-1}{k-1} = k \times \binom{n}{k}$$

  \item For the three following statements, we consider that space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.

  Let $d$ be the line with parametric representation: $\left\{\begin{array}{rl} x &= t + 1 \\ y &= 2t + 1 \\ z &= -t \end{array}\right., t \in \mathbb{R}$.\\
  Let $d'$ be the line with parametric representation: $\left\{\begin{array}{rl} x &= 2t' - 1 \\ y &= -t' + 2 \\ z &= t' + 1 \end{array}\right., t' \in \mathbb{R}$.\\
  Let $P$ be the plane with Cartesian equation: $2x + y - 2z + 18 = 0$.\\
  Let A be the point with coordinates $(-1; -3; 2)$ and B be the point with coordinates $(-5; -5; 6)$. We call the perpendicular bisector plane of segment $[\mathrm{AB}]$ the plane passing through the midpoint of segment $[\mathrm{AB}]$ and perpendicular to the line $(\mathrm{AB})$.

  Statement 3: Point A belongs to line $d$.\\
  Statement 4: Lines $d$ and $d'$ are secant.\\
  Statement 5: Plane $P$ is the perpendicular bisector plane of segment $[\mathrm{AB}]$.
\end{enumerate}