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% --- Basic Settings ---
% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{2BAC.PC/SVT}
\def\examtitle{Série: LES PRIMITIVES, Par Pr: M. ABDELLAOUI}
\def\schoolname{\textbf{Lycée :} Lycée : Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{visit ~~\wsite~~ for more!}}
\def\province{Direction provinciale\newline de Beni Mellal}
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\def\wsite{\href{https://www.mosaid.xyz}{\textcolor{magenta}{\texttt{www.mosaid.xyz}}}}
\def\ddate{\hfill \number\day/\number\month/\number\year~~}

% --- Exam-specific definitions ---

% Exam-specific settings and custom definitions
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% Custom exercise environments (uncomment if needed)
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% Custom commands for this exam (add your own below)
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% Additional packages (uncomment if needed)
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% \usepackage{circuitikz} % For circuit diagrams

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% ----------------------------------------------------------------------
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% --- Custom Header (fr) ---
% Custom header based on default header style 9
% Language: fr
\newcommand{\printheadnine}{%
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\begin{document}



% Exercise 1 (originally 1)
\printexo{1}{}{
Déterminer dans chaque cas les primitives des fonctions suivantes :\\
\begin{minipage}[t]{0.3\textwidth}
\begin{enumerate}
    \item $f(x) = x^8 + x^2$
    \item $f(x) = 3x^2 + 5x + 1$
    \item $f(x) = \dfrac{x^4}{3} - 12x^2 + \dfrac{3}{2}$
    \item $f(x) = x^9 - 3x^2 + 2$
    \item $f(x) = -5x^5 + 3$
    \item $f(x) = -\dfrac{4}{3}x^3 + 6x$
    \item $f(x) = 15x^2 - \dfrac{1}{3}x + 2$
    \item $f(x) = -3x + \dfrac{1}{4}x^3$
\end{enumerate}
\end{minipage}
\begin{minipage}[t]{0.3\textwidth}
  \begin{enumerate}[start=9]
    \item $f(x) = \dfrac{1}{x^2} + 3x$
    \item $f(x) = -\dfrac{2}{3}x + \dfrac{3}{x^2}$
    \item $f(x) = -\dfrac{1}{(x-2)^2}$
    \item $f(x) = \dfrac{3}{(2x-3)^2}$
    \item $f(x) = \dfrac{5}{(-2x+1)^2} + 3$
    \item $f(x) = 2x(x^2 + 3)$
\end{enumerate}
\end{minipage}
\begin{minipage}[t]{0.3\textwidth}
  \begin{enumerate}[start=15]
    \item $f(x) = (x+2)^3$
    \item $f(x) = (3x-2)^4$
    \item $f(x) = x^2(x^3 + 5)^3$
    \item $f(x) = \cos(x)$
    \item $f(x) = \sin(x)$
    \item $f(x) = \cos(3x)$
    \item $f(x) = 1 - \cos(2x)$
    \item $f(x) = \cos\left(3x + \dfrac{\pi}{2}\right)$
\end{enumerate}
\end{minipage}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
Dans chaque cas, déterminer la primitive $F$ de $f$ vérifiant la condition donnée :\\
\begin{minipage}[t]{0.48\textwidth}
\begin{enumerate}
    \item $f(x) = -2x + 4$, et $F(2) = 3$
    \item $f(x) = 8x^3 - 3x$, et $F(1) = 2$
\end{enumerate}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}
  \begin{enumerate}[start=3]
    \item $f(x) = \dfrac{1}{(x+1)^2} + 1$, et $F(0) = 2$
    \item $f(x) = 2\cos(2x) + 2$, et $F\left(\dfrac{\pi}{4}\right) = 1$
\end{enumerate}
\end{minipage}
}

% Exercise 3 (originally 3)
\printexo{3}{}{
Donner les primitives de chacune des fonctions proposées en précisant leurs domaine de définition.\\
\begin{minipage}[t]{0.34\textwidth}
\begin{enumerate}
    \item $f(x) = 2x^2 - 3x + 1$
    \item $f(x) = 3x^4 - 3x^2 + x - 5$
    \item $f(x) = (x-3)^2(x+1)$
    \item $f(x) = (\sqrt{2}x + 3)^2$
    \item $f(x) = (2x+1)\left(x^2 + x + 1\right)^{21}$
\end{enumerate}
\end{minipage}
\begin{minipage}[t]{0.3\textwidth}
  \begin{enumerate}[start=6]
    \item $f(x) = (x+2)^3$
    \item $f(x) = \dfrac{3}{(3x-3)^2}$
    \item $f(x) = \dfrac{2x}{(x^2 + 1)^3}$
    \item $f(x) = \dfrac{3x^2 + 1}{(x^3 + x)^3}$
    \item $f(x) = \dfrac{1 + 2x}{\sqrt{1 + x + x^2}}$
\end{enumerate}
\end{minipage}
\begin{minipage}[t]{0.3\textwidth}
  \begin{enumerate}[start=11]
    \item $f(x) = \dfrac{x + 1}{\sqrt{x^2 + 2x + 21}}$
    \item $f(x) = \sin(x)\cos^3(x)$
    \item $f(x) = \sin^3(x)$
    \item $f(x) = \sqrt[3]{x + 1}$
    \item $f(x) = \dfrac{5}{3}x^5 - \dfrac{3}{4}x^3 + \dfrac{2}{3}x^2 + 2$
\end{enumerate}
\end{minipage}
}

% Exercise 4 (originally 4)
\printexo{4}{}{
Soit $f$ la fonction numérique définie sur $[0, +\infty[$ par :
~$
f(x) = \frac{x^2 + 2x}{(x+1)^2}
$~

\begin{enumerate}
    \item Déterminer les réels $a$ et $b$ tels que :
    ~$
    \forall x \in [0, +\infty[ \quad f(x) = a + \frac{b}{(x+1)^2}
    $~
    \item En déduire les fonctions primitives de la fonction $f$ sur l’intervalle $[0, +\infty[$.
    \item En déduire la fonction primitive $F$ de $f$ sur l’intervalle $[0, +\infty[$ vérifiant :
    ~$
    F(1) = \frac{5}{2}.
    $~
\end{enumerate}
}








\end{document}