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% En-tête
\fancyhead[L]{\textbf{Prof  }}
\fancyhead[C]{\textbf{\Large Integrale d'une Fonction }}
\fancyhead[R]{\textbf{ 2Bac SVT + PC }}

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\fancyfoot[L]{\textbf{Année 2025-2026}} % Texte gauche
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\fancyfoot[R]{\textbf{ Lycée }} % Texte droite
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\begin{document}

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\begin{multicols}{4}
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\section*{Exercice 1 : Polynômes simples et Trigonométrie}
\begin{enumerate}
    \item $\int (3x + 2) \, dx$
    \item $\int (5 - 4x) \, dx$
    \item $\int (x^2 + x + 1) \, dx$
    \item $\int (2x^2 - 3x + 4) \, dx$
    \item $\int (x^3 - 1) \, dx$
    \item $\int (4x^3 + 6x^2) \, dx$
    \item $\int (-x^3 + 2x^2 - x + 7) \, dx$
    \item $\int (\frac{1}{2}x^2 + 3) \, dx$
    \item $\int \cos(x) \, dx$
    \item $\int (2\sin(x) - 3\cos(x)) \, dx$
    \item $\int (\cos(x) + x) \, dx$
    \item $\int (3x^2 + \sin(x)) \, dx$
    \item $\int (ax + b) \, dx$
    \item $\int (7x^3 - 2x + \cos(x)) \, dx$
    \item $\int (10x^2 - 5x + 2) \, dx$
    \item $\int (\sin(x) + \pi) \, dx$
    \item $\int (x^3 + x^2 + x) \, dx$
    \item $\int (6x - \cos(x)) \, dx$
    \item $\int (4 - x^2) \, dx$
\end{enumerate}

\section*{Exercice 2 : Puissances $ax^n + bx^m + \frac{c}{x^p} + dx^r$}
\begin{enumerate}
    \item $\int (x^2 + x^3 + \frac{1}{x^2} + 1) \, dx$
    \item $\int (2x^4 - \frac{3}{x^3}) \, dx$
    \item $\int (x^{1/2} + x^2) \, dx$
    \item $\int (x^{3/2} - 2x^5) \, dx$
    \item $\int (\frac{4}{x^5} + x^{-2}) \, dx$
    \item $\int (3x^{1/3} + \frac{1}{x^4}) \, dx$
    \item $\int (x^{2/3} + x^{1/3} + x) \, dx$
    \item $\int (5x^{-3/2} + x^2) \, dx$
    \item $\int (\frac{1}{x^2} + \frac{1}{x^3} + x^5) \, dx$
    \item $\int (x^{4} + 2x^{1/4}) \, dx$
    \item $\int (\frac{2}{x^{1/2}} + x^3) \, dx$
    \item $\int (x^{-5} + x^{5} + 3) \, dx$
    \item $\int (x^{0.5} + x^{1.5} + \frac{1}{x^4}) \, dx$
    \item $\int (4x^3 + \frac{2}{x^6} - x^{1/3}) \, dx$
    \item $\int (x^{-1/2} + 3x^{3/4}) \, dx$
    \item $\int (\frac{5}{x^2} - x^{2/5}) \, dx$
    \item $\int (x^4 - \frac{1}{x^4}) \, dx$
    \item $\int (2x^{3} + 3x^{-4}) \, dx$
    \item $\int (\frac{1}{x^{0.2}} + x^{1.2}) \, dx$
\end{enumerate}

\section*{Exercice 3 : Forme $\sqrt[n]{ax+b}$}
\begin{enumerate}
    \item $\int \sqrt{x+1} \, dx$
    \item $\int \sqrt{2x+3} \, dx$
    \item $\int \sqrt[3]{x} \, dx$
    \item $\int \sqrt[3]{3x+1} \, dx$
    \item $\int \sqrt{5-x} \, dx$
    \item $\int \sqrt[4]{x+2} \, dx$
    \item $\int \sqrt[5]{2x} \, dx$
    \item $\int \sqrt{4x-2} \, dx$
    \item $\int \sqrt[3]{1-x} \, dx$
    \item $\int \sqrt[6]{x+10} \, dx$
    \item $\int (x+1)^{-1/2} \, dx$
    \item $\int \sqrt{7x+4} \, dx$
    \item $\int \sqrt[3]{5x-2} \, dx$
    \item $\int \sqrt{\frac{1}{2}x + 1} \, dx$
    \item $\int \sqrt[4]{8x+3} \, dx$
    \item $\int \sqrt[5]{x-1} \, dx$
    \item $\int \sqrt[3]{2-3x} \, dx$
    \item $\int \sqrt[7]{x} \, dx$
    \item $\int \sqrt{ax+b} \, dx$
\end{enumerate}

\section*{Exercice 4 : Forme $(ax+b)^n$}
\begin{enumerate}
    \item $\int (x+1)^2 \, dx$
    \item $\int (x-3)^3 \, dx$
    \item $\int (2x+1)^4 \, dx$
    \item $\int (3x-2)^5 \, dx$
    \item $\int (5x+4)^2 \, dx$
    \item $\int (1-x)^6 \, dx$
    \item $\int (2-3x)^3 \, dx$
    \item $\int (\frac{1}{2}x + 1)^4 \, dx$
    \item $\int (4x-7)^2 \, dx$
    \item $\int (x+10)^{10} \, dx$
    \item $\int (2x+3)^{-2} \, dx$
    \item $\int (5-2x)^4 \, dx$
    \item $\int (3x+1)^{1/2} \, dx$
    \item $\int (\frac{x}{3} - 2)^3 \, dx$
    \item $\int (10x-5)^5 \, dx$
    \item $\int (4-x)^7 \, dx$
    \item $\int (6x+2)^3 \, dx$
    \item $\int (ax+b)^n \, dx$
    \item $\int (2x+1)^{-3} \, dx$
\end{enumerate}

\section*{Exercice 5 : Fractions du type $\frac{1}{ax+b}$}
\begin{enumerate}
    \item $\int \frac{1}{x+1} \, dx$
    \item $\int \frac{1}{2x+3} \, dx$
    \item $\int \frac{1}{x-5} \, dx$
    \item $\int \frac{5}{x+4} \, dx$
    \item $\int \frac{1}{3-x} \, dx$
    \item $\int \frac{1}{4x} \, dx$
    \item $\int \frac{2}{3x+1} \, dx$
    \item $\int \frac{-1}{x+2} \, dx$
    \item $\int \frac{10}{2x+5} \, dx$
    \item $\int \frac{1}{x-1} \, dx$
    \item $\int \frac{1}{10-x} \, dx$
    \item $\int \frac{3}{4x-1} \, dx$
    \item $\int \frac{1}{0.5x+2} \, dx$
    \item $\int \frac{4}{2-3x} \, dx$
    \item $\int \frac{1}{ax+b} \, dx$
    \item $\int \frac{1}{\pi x + 1} \, dx$
    \item $\int \frac{1}{7x+3} \, dx$
    \item $\int \frac{-5}{1-x} \, dx$
    \item $\int (\frac{1}{x} + \frac{1}{x+1}) \, dx$
\end{enumerate}

\section*{Exercice 6 : Fonctions Exponentielles $e^{ax+b}$}
\begin{enumerate}
    \item $\int e^x \, dx$
    \item $\int e^{2x} \, dx$
    \item $\int e^{3x+1} \, dx$
    \item $\int e^{-x} \, dx$
    \item $\int 5e^{5x} \, dx$
    \item $\int e^{x/2} \, dx$
    \item $\int e^{-3x+2} \, dx$
    \item $\int (e^x + x) \, dx$
    \item $\int \frac{1}{e^x} \, dx$
    \item $\int 4e^{-2x} \, dx$
    \item $\int (e^{2x} + e^{-2x}) \, dx$
    \item $\int 3e^{x/3} \, dx$
    \item $\int e^{1-x} \, dx$
    \item $\int (e^x + 1)^2 \, dx$
    \item $\int e^{ax+b} \, dx$
    \item $\int \frac{e^x+e^{-x}}{2} \, dx$
    \item $\int e^{x+10} \, dx$
    \item $\int e^{-0.1x} \, dx$
    \item $\int (e^x - e^{-x}) \, dx$
\end{enumerate}

\section*{Exercice 7 : Formes composées $u'u^n$ et $\frac{u'}{u}$}
\begin{enumerate}
    \item $\int 2x(x^2+1)^3 \, dx$
    \item $\int x \sqrt{x^2+1} \, dx$
    \item $\int \frac{2x}{x^2+1} \, dx$
    \item $\int 3x^2(x^3-1)^4 \, dx$
    \item $\int \frac{x}{x^2+4} \, dx$
    \item $\int \sin(x)e^{\cos(x)} \, dx$
    \item $\int \frac{1}{x \ln(x)} \, dx$
    \item $\int \frac{e^x}{\sqrt{e^x+2}} \, dx$
    \item $\int (x+1)(x^2+2x)^3 \, dx$
    \item $\int \frac{\sin(x)}{\cos^2(x)} \, dx$
    \item $\int \frac{e^x}{e^x+1} \, dx$
    \item $\int \cos(x)\sin^2(x) \, dx$
    \item $\int \frac{\cos(x)}{\sin(x)} \, dx$
    \item $\int \frac{1}{x\ln(x)} \, dx$
    \item $\int \frac{1}{\sqrt{x}(\sqrt{x}+1)} \, dx$
    \item $\int \frac{x+1}{x^2+2x+3} \, dx$
    \item $\int x^2 \sqrt{x^3+1} \, dx$
    \item $\int \frac{\ln(x)}{x} \, dx$
    \item $\int \frac{u'(x)}{u(x)} \, dx$
\end{enumerate}
\end{multicols}
\newpage


\begin{multicols}{3}
    \setlength{\columnseprule}{0.9pt}


\section*{Exercice 8 : Mélanges $P(x) + c \cdot e^{ax+b}$ et $P(x) + \sin/\cos$}
\begin{enumerate}
    \item $\int (x^2 + e^{x}) \, dx$
    \item $\int (3x - 2 + e^{2x+1}) \, dx$
    \item $\int (x^3 + 5e^{-x}) \, dx$
    \item $\int (2x^2 - x + 3e^{4x}) \, dx$
    \item $\int (4 - e^{3x-2}) \, dx$
    \item $\int (x + \sin(x)) \, dx$
    \item $\int (x^2 - 2\cos(3x)) \, dx$
    \item $\int (\sin(2x+1) + \cos(x-4)) \, dx$
    \item $\int (3x^2 + \sin(5x) - \cos(2x)) \, dx$
    \item $\int (x^3 + e^{x} + \sin(x)) \, dx$
    \item $\int (10x + 2e^{-2x+3} + \cos(4x)) \, dx$
    \item $\int (ax+b + e^{cx+d}) \, dx$
    \item $\int (\frac{1}{2}x^2 - \sin(x/2)) \, dx$
    \item $\int (e^{x} + \cos(x) - x^4) \, dx$
    \item $\int (3 - 4e^{2x} + \sin(3x+1)) \, dx$
    \item $\int (x^5 - \cos(5x+2)) \, dx$
    \item $\int (2x + 7e^{-x} + \sin(x) - \cos(x)) \, dx$
    \item $\int (\pi x + e^{\pi x} + \sin(\pi)) \, dx$
    \item $\int (x^2 + bx + c + e^{ax}) \, dx$
    \item $\int (x^n + e^{x} + \sin(ax+b)) \, dx$
\end{enumerate}

\section*{Exercice 9 : Synthèse Générale}
\begin{enumerate}
    \item $\int (4x^3 - 2x + 1) \, dx$
    \item $\int \frac{3}{x+2} \, dx$
    \item $\int e^{-5x+2} \, dx$
    \item $\int \sqrt{3x+4} \, dx$
    \item $\int (2x-1)^5 \, dx$
    \item $\int (x^2 + \frac{1}{x^2}) \, dx$
    \item $\int \frac{x}{x^2+1} \, dx$
    \item $\int \sin(4x-3) \, dx$
    \item $\int (e^x + x)^2 \, dx$
    \item $\int \sqrt[3]{2x-1} \, dx$
    \item $\int \frac{1}{4-x} \, dx$
    \item $\int (x\sqrt{x} + \frac{1}{\sqrt{x}}) \, dx$
    \item $\int \cos(x) e^{\sin(x)} \, dx$
    \item $\int (3x+2)^{-4} \, dx$
    \item $\int \frac{e^x}{e^x+5} \, dx$
    \item $\int (x^2 - \sin(x) + e^{2x}) \, dx$
    \item $\int \sqrt[4]{x+1} \, dx$
    \item $\int \frac{2x+3}{x^2+3x+1} \, dx$
    \item $\int (ax+b)^n \, dx$
    \item $\int \frac{1}{\sqrt{2x+5}} \, dx$
    \item $\int (x^3 - \frac{2}{x^3} + \cos(x)) \, dx$
    \item $\int e^{x/2} - e^{-x/2} \, dx$
    \item $\int \frac{\ln(x)}{x} \, dx$
    \item $\int (5x-2)^3 \, dx$
    \item $\int \sqrt{1-x} \, dx$
    \item $\int \frac{1}{3x+7} \, dx$
    \item $\int (x^2+1)x \, dx$
    \item $\int \sin(x) \cos^2(x) \, dx$
    \item $\int 4e^{4x-4} \, dx$
    \item $\int (x+1)^{10} \, dx$
    \item $\int (\frac{1}{x} + e^{-x} + x) \, dx$
    \item $\int \sqrt[5]{x} \, dx$
    \item $\int \frac{e^{2x}}{e^{2x}+3} \, dx$
    \item $\int (2x+1)(x^2+x+1)^2 \, dx$
    \item $\int \cos(10x) \, dx$
    \item $\int (1-3x)^{-2} \, dx$
    \item $\int \frac{1}{x \ln(x)} \, dx$
    \item $\int (e^x + \sin(x) - 7) \, dx$
    \item $\int \sqrt{x+10} \, dx$
    \item $\int \frac{5}{2x-1} \, dx$
    \item $\int x^2(x^3+2)^4 \, dx$
    \item $\int e^{1-x} + \cos(x) \, dx$
    \item $\int (4x-3)^{1/2} \, dx$
    \item $\int \frac{x+1}{(x^2+2x)^2} \, dx$
    \item $\int \sin(x/3) \, dx$
    \item $\int (x^4 + x^2 + 1) \, dx$
    \item $\int \frac{1}{\sqrt[3]{x}} \, dx$
    \item $\int \frac{e^x}{\sqrt{e^x+1}} \, dx$
    \item $\int (2-x)^6 \, dx$
    \item $\int \cos(2x+ \pi/4) \, dx$
    \item $\int (x + \frac{1}{x})^2 \, dx$
    \item $\int \sqrt[n]{x} \, dx$
    \item $\int \frac{3x^2}{x^3-8} \, dx$
    \item $\int e^{ax} \cdot e^b \, dx$
    \item $\int (x-1)(x+1) \, dx$
    \item $\int \frac{\sin(x)}{\cos(x)} \, dx$
    \item $\int (3x+1)^{2/3} \, dx$
    \item $\int e^x + e^{-x} + 2 \, dx$
    \item $\int \frac{1}{(5x+2)^3} \, dx$
    \item $\int (x^2 + \sin(ax+b) + \frac{1}{x}) \, dx$
\end{enumerate}


\end{multicols}

\hrule

\section*{Exercice 10 : Calcul d'intégrales définies (Avec bornes)}
\textit{Calculer la valeur de chaque intégrale suivante :}
 \begin{multicols}{3}
    \setlength{\columnseprule}{0.9pt}

\begin{enumerate}
    \item $\int_{0}^{1} (x^2 - x + 1) \, dx$
    \item $\int_{1}^{2} (3x + 2) \, dx$
    \item $\int_{-1}^{1} x^3 \, dx$
    \item $\int_{0}^{\pi} \sin(x) \, dx$
    \item $\int_{0}^{\pi/2} \cos(x) \, dx$
    \item $\int_{1}^{e} \frac{1}{x} \, dx$
    \item $\int_{0}^{\ln(2)} e^x \, dx$
    \item $\int_{0}^{1} (2x+1)^2 \, dx$
    \item $\int_{1}^{4} \sqrt{x} \, dx$
    \item $\int_{0}^{1} e^{3x} \, dx$
    \item $\int_{2}^{3} \frac{1}{x-1} \, dx$
    \item $\int_{0}^{1} \frac{1}{\sqrt{x+1}} \, dx$
    \item $\int_{0}^{\pi/4} \cos(2x) \, dx$
    \item $\int_{1}^{2} (x + \frac{1}{x^2}) \, dx$
    \item $\int_{0}^{1} (e^x + 1) \, dx$
    \item $\int_{-1}^{0} (4x^3 - 2x) \, dx$
    \item $\int_{0}^{\pi/3} \sin(3x) \, dx$
    \item $\int_{1}^{e^2} \frac{3}{x} \, dx$
    \item $\int_{0}^{4} \sqrt{2x+1} \, dx$
    \item $\int_{0}^{1} \frac{x}{x^2+1} \, dx$
    \item $\int_{1}^{2} \frac{e^x}{e^x-1} \, dx$
    \item $\int_{0}^{2} (x-1)^3 \, dx$
    \item $\int_{1}^{8} \sqrt[3]{x} \, dx$
    \item $\int_{0}^{1} x(x^2+1)^2 \, dx$
    \item $\int_{\pi/6}^{\pi/2} \cos(x) \sin(x) \, dx$
    \item $\int_{1}^{2} \frac{1}{3x+2} \, dx$
    \item $\int_{0}^{\ln(3)} e^{-x} \, dx$
    \item $\int_{0}^{1} (ax+b) \, dx$
    \item $\int_{-2}^{-1} \frac{1}{x^2} \, dx$
    \item $\int_{0}^{\pi} (x + \sin(x)) \, dx$
    \item $\int_{1}^{e} \frac{\ln(x)}{x} \, dx$
    \item $\int_{0}^{1} \frac{1}{(x+1)^3} \, dx$
    \item $\int_{0}^{\pi/4} \frac{\sin(x)}{\cos^2(x)} \, dx$
    \item $\int_{1}^{2} (x^2 - e^{-x}) \, dx$
    \item $\int_{0}^{3} \sqrt{x+1} \, dx$
    \item $\int_{1}^{4} (\frac{1}{\sqrt{x}} + x) \, dx$
    \item $\int_{0}^{1} \frac{e^x+1}{e^x+x} \, dx$
    \item $\int_{0}^{\pi/2} \sin^2(x) \cos(x) \, dx$
    \item $\int_{1}^{2} \frac{2x+1}{x^2+x} \, dx$
    \item $\int_{0}^{1} (1-x)^{10} \, dx$
\end{enumerate}

\end{multicols}

\hrulefill

\section*{Exercice 11 : Vérification de primitives et calcul d'intégrales}
\textit{Pour chaque question : }


\begin{enumerate}
    \item \textit{Montrer que la fonction $F$ est une primitive de $f$ sur l'intervalle donné (en vérifiant que $F'(x) = f(x)$).}
    \item \textit{En déduire la valeur de l'intégrale $I = \int_{a}^{b} f(x) \, dx$.}
\end{enumerate}

\begin{enumerate}
    \item $F(x) = x \ln(x) - x$ ; $f(x) = \ln(x)$ sur $]0; +\infty[$. Calculer $\int_{1}^{e} \ln(x) \, dx$.
    \item $F(x) = \frac{1}{2}e^{x^2}$ ; $f(x) = x e^{x^2}$. Calculer $\int_{0}^{1} x e^{x^2} \, dx$.
    \item $F(x) = \ln(x^2+1)$ ; $f(x) = \frac{2x}{x^2+1}$. Calculer $\int_{0}^{2} \frac{2x}{x^2+1} \, dx$.
    \item $F(x) = (x-1)e^x$ ; $f(x) = x e^x$. Calculer $\int_{0}^{1} x e^x \, dx$.
    \item $F(x) = -\frac{1}{x}$ ; $f(x) = \frac{1}{x^2}$ sur $]0; +\infty[$. Calculer $\int_{1}^{2} \frac{1}{x^2} \, dx$.
    \item $F(x) = \sin(x) - x\cos(x)$ ; $f(x) = x\sin(x)$. Calculer $\int_{0}^{\pi} x\sin(x) \, dx$.
    \item $F(x) = \ln(\ln(x))$ ; $f(x) = \frac{1}{x\ln(x)}$ sur $]1; +\infty[$. Calculer $\int_{e}^{e^2} \frac{1}{x\ln(x)} \, dx$.
    \item $F(x) = \frac{1}{3}(x^2+1)^{3/2}$ ; $f(x) = x\sqrt{x^2+1}$. Calculer $\int_{0}^{\sqrt{3}} x\sqrt{x^2+1} \, dx$.
    \item $F(x) = -\cos(e^x)$ ; $f(x) = e^x \sin(e^x)$. Calculer $\int_{0}^{\ln(\pi)} e^x \sin(e^x) \, dx$.
    \item $F(x) = \frac{x}{2} - \frac{\sin(2x)}{4}$ ; $f(x) = \sin^2(x)$. Calculer $\int_{0}^{\pi/2} \sin^2(x) \, dx$.
    \item $F(x) = \frac{\ln(x)^2}{2}$ ; $f(x) = \frac{\ln(x)}{x}$. Calculer $\int_{1}^{e} \frac{\ln(x)}{x} \, dx$.
    \item $F(x) = \arctan(x)$ ; $f(x) = \frac{1}{1+x^2}$. Calculer $\int_{0}^{1} \frac{1}{1+x^2} \, dx$.
    \item $F(x) = (x^2-2x+2)e^x$ ; $f(x) = x^2 e^x$. Calculer $\int_{0}^{1} x^2 e^x \, dx$.
    \item $F(x) = \frac{1}{a} \ln|ax+b|$ ; $f(x) = \frac{1}{ax+b}$. Calculer $\int_{0}^{1} \frac{1}{2x+1} \, dx$.
    \item $F(x) = \sqrt{x^2+9}$ ; $f(x) = \frac{x}{\sqrt{x^2+9}}$. Calculer $\int_{0}^{4} \frac{x}{\sqrt{x^2+9}} \, dx$.
    \item $F(x) = \frac{2}{3}(x-2)\sqrt{x+1}$ ; $f(x) = \frac{x}{\sqrt{x+1}}$. Calculer $\int_{0}^{3} \frac{x}{\sqrt{x+1}} \, dx$.
    \item $F(x) = e^x(\sin x - \cos x) \frac{1}{2}$ ; $f(x) = e^x \sin x$. Calculer $\int_{0}^{\pi/2} e^x \sin x \, dx$.
    \item $F(x) = \ln|\sin x|$ ; $f(x) = \text{cotan}(x)$. Calculer $\int_{\pi/4}^{\pi/2} \frac{\cos x}{\sin x} \, dx$.
    \item $F(x) = -\frac{1}{\ln x}$ ; $f(x) = \frac{1}{x(\ln x)^2}$. Calculer $\int_{e}^{e^2} \frac{1}{x(\ln x)^2} \, dx$.
    \item $F(x) = \frac{(x+1)^n}{n}$ ; $f(x) = (x+1)^{n-1}$. Calculer $\int_{0}^{1} (x+1)^3 \, dx$.

    \item $F(x) = x \ln(x) - x$ ; $f(x) = \ln(x)$ sur $]0; +\infty[$. Calculer $\int_{1}^{e} \ln(x) \, dx$.
    \item $F(x) = \frac{1}{2}e^{x^2}$ ; $f(x) = x e^{x^2}$. Calculer $\int_{0}^{1} x e^{x^2} \, dx$.
    \item $F(x) = \ln(x^2+1)$ ; $f(x) = \frac{2x}{x^2+1}$. Calculer $\int_{0}^{2} \frac{2x}{x^2+1} \, dx$.
    \item $F(x) = (x-1)e^x$ ; $f(x) = x e^x$. Calculer $\int_{0}^{1} x e^x \, dx$.
    \item $F(x) = -\frac{1}{x}$ ; $f(x) = \frac{1}{x^2}$ sur $]0; +\infty[$. Calculer $\int_{1}^{2} \frac{1}{x^2} \, dx$.
    \item $F(x) = \sin(x) - x\cos(x)$ ; $f(x) = x\sin(x)$. Calculer $\int_{0}^{\pi} x\sin(x) \, dx$.
    \item $F(x) = \ln(\ln(x))$ ; $f(x) = \frac{1}{x\ln(x)}$ sur $]1; +\infty[$. Calculer $\int_{e}^{e^2} \frac{1}{x\ln(x)} \, dx$.
    \item $F(x) = \frac{1}{3}(x^2+1)^{3/2}$ ; $f(x) = x\sqrt{x^2+1}$. Calculer $\int_{0}^{\sqrt{3}} x\sqrt{x^2+1} \, dx$.
    \item $F(x) = -\cos(e^x)$ ; $f(x) = e^x \sin(e^x)$. Calculer $\int_{0}^{\ln(\pi)} e^x \sin(e^x) \, dx$.
    \item $F(x) = \frac{x}{2} - \frac{\sin(2x)}{4}$ ; $f(x) = \sin^2(x)$. Calculer $\int_{0}^{\pi/2} \sin^2(x) \, dx$.
    \item $F(x) = \frac{\ln(x)^2}{2}$ ; $f(x) = \frac{\ln(x)}{x}$. Calculer $\int_{1}^{e} \frac{\ln(x)}{x} \, dx$.
    \item $F(x) = \arctan(x)$ ; $f(x) = \frac{1}{1+x^2}$. Calculer $\int_{0}^{1} \frac{1}{1+x^2} \, dx$.
    \item $F(x) = (x^2-2x+2)e^x$ ; $f(x) = x^2 e^x$. Calculer $\int_{0}^{1} x^2 e^x \, dx$.
    \item $F(x) = \frac{1}{a} \ln|ax+b|$ ; $f(x) = \frac{1}{ax+b}$. Calculer $\int_{0}^{1} \frac{1}{2x+1} \, dx$.
    \item $F(x) = \sqrt{x^2+9}$ ; $f(x) = \frac{x}{\sqrt{x^2+9}}$. Calculer $\int_{0}^{4} \frac{x}{\sqrt{x^2+9}} \, dx$.
    \item $F(x) = \frac{2}{3}(x-2)\sqrt{x+1}$ ; $f(x) = \frac{x}{\sqrt{x+1}}$. Calculer $\int_{0}^{3} \frac{x}{\sqrt{x+1}} \, dx$.
    \item $F(x) = e^x(\sin x - \cos x) \frac{1}{2}$ ; $f(x) = e^x \sin x$. Calculer $\int_{0}^{\pi/2} e^x \sin x \, dx$.
    \item $F(x) = \ln|\sin x|$ ; $f(x) = \frac{cos(x)}{sin(x)}$. Calculer $\int_{\pi/4}^{\pi/2} \frac{\cos x}{\sin x} \, dx$.
    \item $F(x) = -\frac{1}{\ln x}$ ; $f(x) = \frac{1}{x(\ln x)^2}$. Calculer $\int_{e}^{e^2} \frac{1}{x(\ln x)^2} \, dx$.
    \item $F(x) = \frac{(x+3)^n}{n}$ ; $f(x) = (x+3)^{n-1}$. Calculer $\int_{0}^{1} (x+3)^5 \, dx$.

    \item $F(x) = \frac{1}{2} \ln(2x^2+3)$ ; $f(x) = \frac{2x}{2x^2+3}$. Calculer $\int_{0}^{1} \frac{2x}{2x^2+3} \, dx$.
    \item $F(x) = (x^2-2) \sin(x) + 2x \cos(x)$ ; $f(x) = x^2 \cos(x)$. Calculer $\int_{0}^{\pi/2} x^2 \cos(x) \, dx$.
    \item $F(x) = \ln(x+e^x)$ ; $f(x) = \frac{1+e^x}{x+e^x}$. Calculer $\int_{1}^{2} \frac{1+e^x}{x+e^x} \, dx$.
    \item $F(x) = -\frac{1}{2} \cos(x^2)$ ; $f(x) = x \sin(x^2)$. Calculer $\int_{0}^{\sqrt{\pi}} x \sin(x^2) \, dx$.
    \item $F(x) = \frac{x-1}{x+1}$ ; $f(x) = \frac{2}{(x+1)^2}$. Calculer $\int_{0}^{1} \frac{2}{(x+1)^2} \, dx$.
    \item $F(x) = \frac{2}{3}(x+1) \sqrt{x+1}$ ; $f(x) = \sqrt{x+1}$. Calculer $\int_{0}^{3} \sqrt{x+1} \, dx$.
    \item $F(x) = e^x \ln(x) + \int \frac{e^x}{x} dx$ (Astuce) ; $f(x) = e^x (\ln x + \frac{1}{x})$. Calculer $\int_{1}^{e} e^x (\ln x + \frac{1}{x}) \, dx$.
    \item $F(x) = \frac{1}{2}(\ln(x+1))^2$ ; $f(x) = \frac{\ln(x+1)}{x+1}$. Calculer $\int_{0}^{e-1} \frac{\ln(x+1)}{x+1} \, dx$.
    \item $F(x) = \frac{\sin^4(x)}{4}$ ; $f(x) = \cos(x) \sin^3(x)$. Calculer $\int_{0}^{\pi/2} \cos(x) \sin^3(x) \, dx$.
    \item $F(x) = x - 2\sqrt{x}$ ; $f(x) = 1 - \frac{1}{\sqrt{x}}$. Calculer $\int_{1}^{4} (1 - \frac{1}{\sqrt{x}}) \, dx$.
    \item $F(x) = \frac{e^{2x}}{2} + e^x + x$ ; $f(x) = (e^x+1)^2$. Calculer $\int_{0}^{1} (e^x+1)^2 \, dx$.
    \item $F(x) = \ln|\cos x|$ ; $f(x) = -\tan(x)$. Calculer $\int_{0}^{\pi/3} \tan(x) \, dx$.
    \item $F(x) = \frac{-1}{x^2+x+1}$ ; $f(x) = \frac{2x+1}{(x^2+x+1)^2}$. Calculer $\int_{0}^{1} \frac{2x+1}{(x^2+x+1)^2} \, dx$.
    \item $F(x) = \frac{1}{2} e^{2x-1}$ ; $f(x) = e^{2x-1}$. Calculer $\int_{1/2}^{1} e^{2x-1} \, dx$.
    \item $F(x) = (x-2)\sqrt{2x+1} \cdot \frac{1}{3}$ (approx) ; $f(x) = \frac{x}{\sqrt{2x+1}}$. Calculer $\int_{0}^{4} \frac{x}{\sqrt{2x+1}} \, dx$.
    \item $F(x) = \sin(\ln x)$ ; $f(x) = \frac{\cos(\ln x)}{x}$. Calculer $\int_{1}^{e^{\pi/2}} \frac{\cos(\ln x)}{x} \, dx$.
    \item $F(x) = \frac{1}{n+1} \ln(x^{n+1}+1)$ ; $f(x) = \frac{x^n}{x^{n+1}+1}$. Calculer $\int_{0}^{1} \frac{x^2}{x^3+1} \, dx$.
    \item $F(x) = \ln(x^2+4)$ ; $f(x) = \frac{2x}{x^2+4}$. Calculer $\int_{0}^{2} \frac{2x}{x^2+4} \, dx$.
    \item $F(x) = \frac{1}{2} x^2 - \frac{1}{x}$ ; $f(x) = x + \frac{1}{x^2}$. Calculer $\int_{1}^{2} (x + \frac{1}{x^2}) \, dx$.
    \item $F(x) = e^x(x-1)$ ; $f(x) = x e^x$. Calculer $\int_{-1}^{1} x e^x \, dx$.

    \item $F(x) = \ln|x^2-1|$ ; $f(x) = \frac{2x}{x^2-1}$. Calculer $\int_{2}^{3} \frac{2x}{x^2-1} \, dx$.
    \item $F(x) = \frac{2}{3} (x-2)\sqrt{x+1}$ ; $f(x) = \frac{x}{\sqrt{x+1}}$. Calculer $\int_{3}^{8} \frac{x}{\sqrt{x+1}} \, dx$.
    \item $F(x) = \frac{-1}{\sin x}$ ; $f(x) = \frac{\cos x}{\sin^2 x}$. Calculer $\int_{\pi/4}^{\pi/2} \frac{\cos x}{\sin^2 x} \, dx$.
    \item $F(x) = \frac{1}{2} \ln(1+e^{2x})$ ; $f(x) = \frac{e^{2x}}{1+e^{2x}}$. Calculer $\int_{0}^{1} \frac{e^{2x}}{1+e^{2x}} \, dx$.
    \item $F(x) = \frac{x^3}{3} \ln x - \frac{x^3}{9}$ ; $f(x) = x^2 \ln x$. Calculer $\int_{1}^{e} x^2 \ln x \, dx$.
    \item $F(x) = \sqrt{e^x+1}$ ; $f(x) = \frac{e^x}{2\sqrt{e^x+1}}$. Calculer $\int_{0}^{\ln 3} \frac{e^x}{2\sqrt{e^x+1}} \, dx$.
    \item $F(x) = \frac{1}{2} (\sin x + \cos x) e^x$ ; $f(x) = e^x \cos x$. Calculer $\int_{0}^{\pi/2} e^x \cos x \, dx$.
    \item $F(x) = \ln|x + \sqrt{x^2+1}|$ ; $f(x) = \frac{1}{\sqrt{x^2+1}}$. Calculer $\int_{0}^{1} \frac{1}{\sqrt{x^2+1}} \, dx$.
    \item $F(x) = \frac{-1}{x \ln x}$ (Faux) $\to F(x) = \ln|\ln x|$ ; $f(x) = \frac{1}{x \ln x}$. Calculer $\int_{e}^{e^2} \frac{1}{x \ln x} \, dx$.
    \item $F(x) = \frac{(ax+b)^{n+1}}{a(n+1)}$ ; $f(x) = (ax+b)^n$. Calculer $\int_{0}^{1} (3x+1)^2 \, dx$.
    \item $F(x) = x \arcsin x + \sqrt{1-x^2}$ ; $f(x) = \arcsin x$. Calculer $\int_{0}^{1} \arcsin x \, dx$.
    \item $F(x) = \ln|x^2-5x+6|$ ; $f(x) = \frac{2x-5}{x^2-5x+6}$. Calculer $\int_{4}^{5} \frac{2x-5}{x^2-5x+6} \, dx$.
    \item $F(x) = \frac{1}{2} \tan^2 x$ ; $f(x) = \frac{\tan x}{\cos^2 x}$. Calculer $\int_{0}^{\pi/4} \frac{\tan x}{\cos^2 x} \, dx$.
    \item $F(x) = \frac{-1}{e^x+1}$ ; $f(x) = \frac{e^x}{(e^x+1)^2}$. Calculer $\int_{0}^{1} \frac{e^x}{(e^x+1)^2} \, dx$.
    \item $F(x) = \frac{x}{2} + \frac{\sin(2x)}{4}$ ; $f(x) = \cos^2 x$. Calculer $\int_{0}^{\pi} \cos^2 x \, dx$.
    \item $F(x) = (x^2+1)e^x$ ; $f(x) = (x+1)^2 e^x$. Calculer $\int_{0}^{1} (x+1)^2 e^x \, dx$.
    \item $F(x) = \frac{2}{3} (x-b/a) \sqrt{ax+b} \cdot \frac{1}{a}$ (forme) ; $f(x) = \frac{x}{\sqrt{ax+b}}$. Calculer $\int_{1}^{2} \frac{x}{\sqrt{x+1}} \, dx$.
    \item $F(x) = \ln|1+\tan x|$ ; $f(x) = \frac{1+\tan^2 x}{1+\tan x}$. Calculer $\int_{0}^{\pi/4} \frac{1+\tan^2 x}{1+\tan x} \, dx$.
    \item $F(x) = \frac{1}{2} \ln^2(x^2+1)$ ; $f(x) = \frac{2x \ln(x^2+1)}{x^2+1}$. Calculer $\int_{0}^{1} \frac{2x \ln(x^2+1)}{x^2+1} \, dx$.
    \item $F(x) = \ln|\sin x + \cos x|$ ; $f(x) = \frac{\cos x - \sin x}{\sin x + \cos x}$. Calculer $\int_{0}^{\pi/4} \frac{\cos x - \sin x}{\sin x + \cos x} \, dx$.


\end{enumerate}

    \section*{Exercice 12 : Interprétation graphique et calcul d'aires}


 \begin{multicols}{2}
    \setlength{\columnseprule}{0.9pt}

\textit{Dans chaque cas, on considère une fonction $f$ continue sur un intervalle $[a ; b]$. On note $\mathcal{A}$ l'aire de la surface délimitée par la courbe de $f$, l'axe des abscisses et les droites d'équations $x=a$ et $x=b$. Calculer la mesure de cette surface hachurée (en unités d'aire).}

\begin{enumerate}
    \item $f_1(x) = e^{-x}$ entre $x = 0$ et $x = 1$.
    \item $f_2(x) = \frac{1}{x}$ entre $x = 1$ et $x = e$.
    \item $f_3(x) = x^2$ entre $x = 0$ et $x = 2$.
    \item $f_4(x) = \sin(x)$ entre $x = 0$ et $x = \pi$.
    \item $f_5(x) = \sqrt{x}$ entre $x = 1$ et $x = 4$.
    \item $f_6(x) = \frac{1}{x^2}$ entre $x = 1$ et $x = 3$.
    \item $f_7(x) = 2x + 1$ entre $x = 0$ et $x = 2$.
    \item $f_8(x) = e^x$ entre $x = -1$ et $x = 1$.
    \item $f_9(x) = \cos(x)$ entre $x = 0$ et $x = \pi/2$.
    \item $f_{10}(x) = \frac{2x}{x^2+1}$ entre $x = 0$ et $x = 1$.
    \item $f_{11}(x) = (x-1)^2$ entre $x = 1$ et $x = 3$.
    \item $f_{12}(x) = \ln(x)$ entre $x = 1$ et $x = e$ (utiliser la primitive $x\ln x - x$).
    \item $f_{13}(x) = \frac{1}{x+1}$ entre $x = 0$ et $x = 1$.
    \item $f_{14}(x) = 4 - x^2$ entre $x = 0$ et $x = 2$.
    \item $f_{15}(x) = e^{2x}$ entre $x = 0$ et $x = \ln 2$.
    \item $f_{16}(x) = \frac{1}{\sqrt{x}}$ entre $x = 1$ et $x = 4$.
    \item $f_{17}(x) = x^3$ entre $x = 0$ et $x = 1$.
    \item $f_{18}(x) = 1 + \cos(x)$ entre $x = 0$ et $x = \pi$.
    \item $f_{19}(x) = \frac{e^x}{e^x+1}$ entre $x = 0$ et $x = 1$.
    \item $f_{20}(x) = \sqrt{2x+1}$ entre $x = 0$ et $x = 4$.



\end{enumerate}
\end{multicols}

\section*{Exercice 12 - Série B : Aires et Représentations Graphiques}

\textit{Pour chaque cas, esquisser la courbe représentative de la fonction, hachurer la surface délimitée par la courbe, l'axe des abscisses et les droites verticales données, puis calculer l'aire.}

 \begin{multicols}{2}
    \setlength{\columnseprule}{0.9pt}

\begin{enumerate}
    \item $f_{21}(x) = 1 + x^2$ entre $x = -1$ et $x = 1$.
    \item $f_{22}(x) = \frac{1}{x+2}$ entre $x = 0$ et $x = 2$.
    \item $f_{23}(x) = e^{x/2}$ entre $x = 0$ et $x = 2$.
    \item $f_{24}(x) = 2\sin(x)$ entre $x = 0$ et $x = \pi/2$.
    \item $f_{25}(x) = \frac{1}{\sqrt{2x+4}}$ entre $x = 0$ et $x = 6$.
    \item $f_{26}(x) = x^2 - 2x + 1$ entre $x = 0$ et $x = 2$.
    \item $f_{27}(x) = \frac{1}{x^2+1}$ entre $x = 0$ et $x = 1$.
    \item $f_{28}(x) = 3x^2$ entre $x = -1$ et $x = 0$.
    \item $f_{29}(x) = \ln(x+1)$ entre $x = 0$ et $x = e-1$.
    \item $f_{30}(x) = e^{-2x}$ entre $x = 0$ et $x = 1$.
    \item $f_{31}(x) = \sqrt[3]{x}$ entre $x = 1$ et $x = 8$.
    \item $f_{32}(x) = \cos^2(x)\sin(x)$ entre $x = 0$ et $x = \pi/2$.
    \item $f_{33}(x) = \frac{e^x}{e^x+2}$ entre $x = 0$ et $x = \ln 2$.
    \item $f_{34}(x) = 5 - x$ entre $x = 1$ et $x = 4$.
    \item $f_{35}(x) = \frac{1}{x}$ entre $x = e$ et $x = e^2$.
    \item $f_{36}(x) = \sin(2x)$ entre $x = 0$ et $x = \pi/4$.
    \item $f_{37}(x) = (x+1)^3$ entre $x = 0$ et $x = 1$.
    \item $f_{38}(x) = \frac{x}{\sqrt{x^2+1}}$ entre $x = 0$ et $x = \sqrt{3}$.
    \item $f_{39}(x) = 2 - e^{-x}$ entre $x = 0$ et $x = 2$.
    \item $f_{40}(x) = \pi \sin(\pi x)$ entre $x = 0$ et $x = 1$.
\end{enumerate}
\end{multicols}


\section*{Exercice 13 : Calcul d'aires à partir de graphiques}
 \begin{multicols}{2}
    \setlength{\columnseprule}{0.9pt}

\textit{Pour chaque graphique ci-dessous, la fonction est de la forme $f(x) = \frac{ax+b}{cx+d}$. Déterminez l'expression de la fonction à partir du graphique (ou utilisez celle donnée), puis calculez l'aire de la surface hachurée entre les bornes indiquées.}

% --- CAS 1 : f(x) = (x+1)/(x-2) ---
\subsection*{1) Aire sous $f(x) = \frac{x+1}{x-2}$ entre $x=3$ et $x=5$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=2.1:7, % On commence après l'asymptote x=2
            samples=100,
            axis lines=middle,
            xlabel=$x$, ylabel=$f(x)$,
            ymin=-2, ymax=6,
            xtick={0,1,2,3,4,5,6},
            ytick={-2,0,2,4,6},
            grid=both,
            title={Graphique de $f(x) = \frac{x+1}{x-2}$},
        ]
        % Surface hachurée
        \addplot [fill=blue!20, domain=3:5] {(x+1)/(x-2)} \closedcycle;

        % Courbe
        \addplot [color=red, thick] {(x+1)/(x-2)};

        % Asymptote verticale x = 2
        \draw[dashed, blue] (axis cs:2,-2) -- (axis cs:2,6);
        % Asymptote horizontale y = 1
        \draw[dashed, green!70!black] (axis cs:0,1) -- (axis cs:7,1);
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- CAS 2 : f(x) = (2x)/(x+1) ---
\subsection*{2) Aire sous $f(x) = \frac{2x}{x+1}$ entre $x=0$ et $x=2$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-0.5:5,
            samples=100,
            axis lines=middle,
            xlabel=$x$, ylabel=$g(x)$,
            ymin=-1, ymax=4,
            grid=both,
            title={Graphique de $g(x) = \frac{2x}{x+1}$},
        ]
        \addplot [fill=orange!20, domain=0:2] {(2*x)/(x+1)} \closedcycle;
        \addplot [color=red, thick] {(2*x)/(x+1)};
        \draw[dashed, blue] (axis cs:-1,-1) -- (axis cs:-1,4);
        \draw[dashed, green!70!black] (axis cs:-0.5,2) -- (axis cs:5,2);
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- CAS 3 : f(x) = (x-4)/(x-5) ---
\subsection*{3) Aire sous $h(x) = \frac{x-4}{x-5}$ entre $x=0$ et $x=3$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-1:4.5,
            samples=100,
            axis lines=middle,
            grid=both,
            title={Graphique de $h(x) = \frac{x-4}{x-5}$},
        ]
        \addplot [fill=green!20, domain=0:3] {(x-4)/(x-5)} \closedcycle;
        \addplot [color=red, thick] {(x-4)/(x-5)};
        \draw[dashed, blue] (axis cs:5,-1) -- (axis cs:5,3);
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- CAS 4 : f(x) = (3x+6)/(x+4) ---
\subsection*{4) Aire sous $k(x) = \frac{3x+6}{x+4}$ entre $x=-2$ et $x=2$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-3.5:4,
            samples=100,
            axis lines=middle,
            grid=both,
            title={Graphique de $k(x) = \frac{3x+6}{x+4}$},
        ]
        \addplot [fill=red!20, domain=-2:2] {(3*x+6)/(x+4)} \closedcycle;
        \addplot [color=red, thick] {(3*x+6)/(x+4)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- CAS 5 : f(x) = (x)/(2x+2) ---
\subsection*{5) Aire sous $m(x) = \frac{x}{2x+2}$ entre $x=1$ et $x=4$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=0:5,
            samples=100,
            axis lines=middle,
            grid=both,
            title={Graphique de $m(x) = \frac{x}{2x+2}$},
        ]
        \addplot [fill=purple!20, domain=1:4] {(x)/(2*x+2)} \closedcycle;
        \addplot [color=red, thick] {(x)/(2*x+2)};
        \draw[dashed, green!70!black] (axis cs:0,0.5) -- (axis cs:5,0.5);
        \end{axis}
    \end{tikzpicture}
\end{center}

\section*{Exercice 14 : Aires sous courbes variées }

\textit{Pour chaque graphique, calculez l'aire de la surface colorée délimitée par la courbe, l'axe des abscisses et les bornes indiquées.}

% --- g1 : x² - 5x + 4 ---
\subsection*{6) Aire sous $g_1(x) = x^2 - 5x + 4$ entre $x=0$ et $x=1$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-0.5:5, samples=100,
            axis lines=middle, xlabel=$x$, ylabel=$y$,
            ymin=-3, ymax=5, grid=both,
            title={Graphique de $g_1(x) = x^2 - 5x + 4$}
        ]
        \addplot [fill=blue!30, domain=0:1] {x^2 - 5*x + 4} \closedcycle;
        \addplot [color=red, thick] {x^2 - 5*x + 4};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g2 : 4x + 3 ---
\subsection*{7) Aire sous $g_2(x) = 4x + 3$ entre $x=0$ et $x=2$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-1:3, samples=50,
            axis lines=middle, xlabel=$x$, ylabel=$y$,
            ymin=0, ymax=12, grid=both,
            title={Graphique de $g_2(x) = 4x + 3$}
        ]
        \addplot [fill=green!30, domain=0:2] {4*x + 3} \closedcycle;
        \addplot [color=red, thick] {4*x + 3};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g3 : e^(-x) ---
\subsection*{8) Aire sous $g_3(x) = e^{-x}$ entre $x=0$ et $x=2$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-0.5:3, samples=100,
            axis lines=middle, xlabel=$x$, ylabel=$y$,
            ymin=0, ymax=1.5, grid=both,
            title={Graphique de $g_3(x) = e^{-x}$}
        ]
        \addplot [fill=red!20, domain=0:2] {exp(-x)} \closedcycle;
        \addplot [color=red, thick] {exp(-x)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g4 : sqrt(x) ---
\subsection*{9) Aire sous $g_4(x) = \sqrt{x}$ entre $x=0$ et $x=4$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=0:5, samples=100,
            axis lines=middle, xlabel=$x$, ylabel=$y$,
            ymin=0, ymax=3, grid=both,
            title={Graphique de $g_4(x) = \sqrt{x}$}
        ]
        \addplot [fill=orange!30, domain=0:4] {sqrt(x)} \closedcycle;
        \addplot [color=red, thick] {sqrt(x)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g5 : 2/x² ---
\subsection*{10) Aire sous $g_5(x) = \frac{2}{x^2}$ entre $x=1$ et $x=3$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=0.5:4, samples=100,
            axis lines=middle, xlabel=$x$, ylabel=$y$,
            ymin=0, ymax=3, grid=both,
            title={Graphique de $g_5(x) = \frac{2}{x^2}$}
        ]
        \addplot [fill=purple!20, domain=1:3] {2/(x^2)} \closedcycle;
        \addplot [color=red, thick] {2/(x^2)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g6 : sin(x) ---
\subsection*{11) Aire sous $g_6(x) = \sin(x)$ entre $x=0$ et $x=\pi$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=0:pi, samples=100,
            axis lines=middle, xlabel=$x$, ylabel=$y$,
            ymin=0, ymax=1.2, grid=both,
            xtick={0, 1.57, 3.14},
            xticklabels={0, $\frac{\pi}{2}$, $\pi$},
            title={Graphique de $g_6(x) = \sin(x)$}
        ]
        \addplot [fill=yellow!30, domain=0:pi] {sin(deg(x))} \closedcycle;
        \addplot [color=red, thick] {sin(deg(x))};
        \end{axis}
    \end{tikzpicture}
\end{center}


\section*{Exercice 15  : Variété de Courbes}

\textit{Calculez l'aire de la surface hachurée pour chaque cas ci-dessous en utilisant les outils du calcul intégral.}

% --- g12 : x*exp(-x) ---
\subsection*{12) Aire sous $g_{12}(x) = x e^{-x}$ entre $x=0$ et $x=2$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-0.5:4, samples=100,
            axis lines=middle, xlabel=$x$, ylabel=$y$,
            ymin=0, ymax=1, grid=both,
            title={Graphique de $g_{12}(x) = x e^{-x}$}
        ]
        \addplot [fill=teal!40, domain=0:2] {x*exp(-x)} \closedcycle;
        \addplot [color=teal, ultra thick] {x*exp(-x)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g13 : 1/(x+1) ---
\subsection*{13) Aire sous $g_{13}(x) = \frac{1}{x+1}$ entre $x=0$ et $x=3$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-0.5:4, samples=100,
            axis lines=middle, grid=both,
            title={Graphique de $g_{13}(x) = \frac{1}{x+1}$}
        ]
        \addplot [fill=magenta!30, domain=0:3] {1/(x+1)} \closedcycle;
        \addplot [color=magenta, ultra thick] {1/(x+1)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g14 : x^3 ---
\subsection*{14) Aire sous $g_{14}(x) = x^3$ entre $x=0$ et $x=1.5$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-0.5:2, samples=100,
            axis lines=middle, grid=both,
            title={Graphique de $g_{14}(x) = x^3$}
        ]
        \addplot [fill=cyan!30, domain=0:1.5] {x^3} \closedcycle;
        \addplot [color=cyan!70!black, ultra thick] {x^3};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g15 : ln(x) ---
\subsection*{15) Aire sous $g_{15}(x) = \ln(x)$ entre $x=1$ et $x=e$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=0.5:3.5, samples=100,
            axis lines=middle, grid=both,
            xtick={1, 2.718}, xticklabels={1, $e$},
            title={Graphique de $g_{15}(x) = \ln(x)$}
        ]
        \addplot [fill=olive!40, domain=1:2.718] {ln(x)} \closedcycle;
        \addplot [color=olive, ultra thick] {ln(x)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g16 : 4 - x^2 ---
\subsection*{16) Aire sous $g_{16}(x) = 4 - x^2$ entre $x=-2$ et $x=2$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-3:3, samples=100,
            axis lines=middle, grid=both,
            title={Graphique de $g_{16}(x) = 4 - x^2$}
        ]
        \addplot [fill=violet!30, domain=-2:2] {4-x^2} \closedcycle;
        \addplot [color=violet, ultra thick] {4-x^2};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g17 : cos(x) ---
\subsection*{17) Aire sous $g_{17}(x) = \cos(x)$ entre $x=-\pi/2$ et $x=\pi/2$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-2:2, samples=100,
            axis lines=middle, grid=both,
            xtick={-1.57, 0, 1.57},
            xticklabels={$-\frac{\pi}{2}$, 0, $\frac{\pi}{2}$},
            title={Graphique de $g_{17}(x) = \cos(x)$}
        ]
        \addplot [fill=brown!30, domain=-1.57:1.57] {cos(deg(x))} \closedcycle;
        \addplot [color=brown, ultra thick] {cos(deg(x))};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g18 : exp(x) + 1 ---
\subsection*{18) Aire sous $g_{18}(x) = e^x + 1$ entre $x=-1$ et $x=1$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-2:2, samples=100,
            axis lines=middle, grid=both,
            title={Graphique de $g_{18}(x) = e^x + 1$}
        ]
        \addplot [fill=pink!50, domain=-1:1] {exp(x)+1} \closedcycle;
        \addplot [color=red!70!black, ultra thick] {exp(x)+1};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g19 : 1/sqrt(x) ---
\subsection*{19) Aire sous $g_{19}(x) = \frac{1}{\sqrt{x}}$ entre $x=1$ et $x=4$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=0.5:5, samples=100,
            axis lines=middle, grid=both,
            title={Graphique de $g_{19}(x) = \frac{1}{\sqrt{x}}$}
        ]
        \addplot [fill=lime!30, domain=1:4] {1/sqrt(x)} \closedcycle;
        \addplot [color=lime!60!black, ultra thick] {1/sqrt(x)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g20 : 2x / (x^2+1) ---
\subsection*{20) Aire sous $g_{20}(x) = \frac{2x}{x^2+1}$ entre $x=0$ et $x=3$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-1:4, samples=100,
            axis lines=middle, grid=both,
            title={Graphique de $g_{20}(x) = \frac{2x}{x^2+1}$}
        ]
        \addplot [fill=darkgray!20, domain=0:3] {(2*x)/(x^2+1)} \closedcycle;
        \addplot [color=black, ultra thick] {(2*x)/(x^2+1)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- g21 : 3 ---
\subsection*{21) Aire sous la fonction constante $g_{21}(x) = 3$ entre $x=1$ et $x=5$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=0:6, samples=10,
            axis lines=middle, grid=both,
            ymin=0, ymax=4,
            title={Graphique de la fonction constante}
        ]
        \addplot [fill=blue!10, domain=1:5] {3} \closedcycle;
        \addplot [color=blue, ultra thick] {3};
        \end{axis}
    \end{tikzpicture}
\end{center}


\end{multicols}


\section*{Exercice 16 : Aire comprise entre deux courbes}

\textit{Pour chaque cas, identifiez la fonction dominante (celle du dessus), déterminez les points d'intersection si nécessaire, et calculez l'aire de la surface comprise entre les deux courbes.}

 \begin{multicols}{2}
    \setlength{\columnseprule}{0.9pt}


    % --- Cas 10 : e^(x-1) et 1 ---
\subsection*{1) Entre $f(x) = e^{x-1}$ et $g(x) = 1$ sur $[0 ; 1]$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-0.5:1.5, samples=100,
            axis lines=middle, grid=both,
            title={Région entre $y=1$ et $y=e^{x-1}$}
        ]
        \addplot [fill=violet!20, domain=0:1] {1} \closedcycle;
        \addplot [fill=white, domain=0:1] {exp(x-1)} \closedcycle;

        \addplot [color=black, thick] {1};
        \addplot [color=violet, ultra thick] {exp(x-1)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- Cas 1 : Parabole et Droite ---
\subsection*{1) Entre $f(x) = x^2$ et $g(x) = x+2$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-2.5:3, samples=100,
            axis lines=middle, grid=both,
            title={Région entre $y=x^2$ et $y=x+2$}
        ]
        % Remplissage entre les deux courbes
        \addplot [fill=cyan!20, domain=-1:2] {x+2} \closedcycle;
        \addplot [fill=white, domain=-1:2] {x^2} \closedcycle; % On "vide" le dessous

        \addplot [color=blue, ultra thick] {x^2};
        \addplot [color=red, ultra thick] {x+2};
        \node at (axis cs:1.5,4) {$g(x)$};
        \node at (axis cs:-1.8,3) {$f(x)$};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- Cas 2 : Sinus et Cosinus ---
\subsection*{2) Entre $f(x) = \sin(x)$ et $g(x) = \cos(x)$ sur $[0 ; \pi/4]$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=0:1.5, samples=100,
            axis lines=middle, grid=both,
            title={Région entre $\sin(x)$ et $\cos(x)$}
        ]
        \addplot [fill=orange!30, domain=0:0.785] {cos(deg(x))} \closedcycle;
        \addplot [fill=white, domain=0:0.785] {sin(deg(x))} \closedcycle;

        \addplot [color=orange, thick] {cos(deg(x))};
        \addplot [color=purple, thick] {sin(deg(x))};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- Cas 3 : Exponentielle et Droite ---
\subsection*{3) Entre $f(x) = e^x$ et $g(x) = x+1$ sur $[0 ; 1]$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-0.5:1.5, samples=100,
            axis lines=middle, grid=both,
            title={Région entre $e^x$ et $x+1$}
        ]
        \addplot [fill=green!20, domain=0:1] {exp(x)} \closedcycle;
        \addplot [fill=white, domain=0:1] {x+1} \closedcycle;

        \addplot [color=green!60!black, ultra thick] {exp(x)};
        \addplot [color=black, thick] {x+1};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- Cas 4 : Deux Paraboles ---
\subsection*{4) Entre $f(x) = 4-x^2$ et $g(x) = x^2-4$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-3:3, samples=100,
            axis lines=middle, grid=both,
            title={Région entre deux paraboles}
        ]
        \addplot [fill=magenta!20, domain=-2:2] {4-x^2} \closedcycle;
        \addplot [fill=magenta!20, domain=-2:2] {x^2-4} \closedcycle;

        \addplot [color=red, thick] {4-x^2};
        \addplot [color=blue, thick] {x^2-4};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- Cas 5 : Racine et Carré ---
\subsection*{5) Entre $f(x) = \sqrt{x}$ et $g(x) = x^2$ sur $[0 ; 1]$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=0:1.2, samples=100,
            axis lines=middle, grid=both,
            title={L'oeil de Newton : $\sqrt{x}$ et $x^2$}
        ]
        \addplot [fill=yellow!40, domain=0:1] {sqrt(x)} \closedcycle;
        \addplot [fill=white, domain=0:1] {x^2} \closedcycle;

        \addplot [color=orange, thick] {sqrt(x)};
        \addplot [color=brown, thick] {x^2};
        \end{axis}
    \end{tikzpicture}
\end{center}


\section*{Exercice 17 - Série B : Aires entre courbes (Exponentielles et Logarithmes)}

\textit{Déterminez l'aire de la surface comprise entre les deux courbes données sur l'intervalle indiqué.}

% --- Cas 6 : e^x et e^-x ---
\subsection*{6) Entre $f(x) = e^x$ et $g(x) = e^{-x}$ sur $[0 ; 1]$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-0.5:1.5, samples=100,
            axis lines=middle, grid=both,
            title={Région entre $e^x$ et $e^{-x}$}
        ]
        \addplot [fill=cyan!30, domain=0:1] {exp(x)} \closedcycle;
        \addplot [fill=white, domain=0:1] {exp(-x)} \closedcycle;

        \addplot [color=blue, ultra thick] {exp(x)};
        \addplot [color=red, ultra thick] {exp(-x)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- Cas 7 : ln(x) et l'axe (ou une droite) ---
\subsection*{7) Entre $f(x) = \ln(x)$ et $g(x) = 1$ sur $[1 ; e]$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=0.5:3.5, samples=100,
            axis lines=middle, grid=both,
            xtick={1, 2.718}, xticklabels={1, $e$},
            title={Région entre $y=1$ et $y=\ln(x)$}
        ]
        \addplot [fill=olive!30, domain=1:2.718] {1} \closedcycle;
        \addplot [fill=white, domain=1:2.718] {ln(x)} \closedcycle;

        \addplot [color=black, ultra thick] {1};
        \addplot [color=olive, ultra thick] {ln(x)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- Cas 8 : e^(2x) et e^x ---
\subsection*{8) Entre $f(x) = e^{2x}$ et $g(x) = e^x$ sur $[0 ; \ln 2]$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=-0.2:1, samples=100,
            axis lines=middle, grid=both,
            title={Région entre $e^{2x}$ et $e^x$}
        ]
        \addplot [fill=magenta!20, domain=0:0.693] {exp(2*x)} \closedcycle;
        \addplot [fill=white, domain=0:0.693] {exp(x)} \closedcycle;

        \addplot [color=magenta, ultra thick] {exp(2*x)};
        \addplot [color=blue, ultra thick] {exp(x)};
        \end{axis}
    \end{tikzpicture}
\end{center}

% --- Cas 9 : x et ln(x) ---
\subsection*{9) Entre $f(x) = x$ et $g(x) = \ln(x)$ sur $[1 ; 2]$}
\begin{center}
    \begin{tikzpicture}
        \begin{axis}[
            domain=0.5:2.5, samples=100,
            axis lines=middle, grid=both,
            title={Région entre $y=x$ et $y=\ln(x)$}
        ]
        \addplot [fill=teal!30, domain=1:2] {x} \closedcycle;
        \addplot [fill=white, domain=1:2] {ln(x)} \closedcycle;

        \addplot [color=black, thick] {x};
        \addplot [color=teal, ultra thick] {ln(x)};
        \end{axis}
    \end{tikzpicture}
\end{center}



\end{multicols}

\section*{Exercice 18 : Intégration par parties (IPP)}
\textit{Calculer les intégrales suivantes en utilisant la méthode d'intégration par parties :}

\subsection*{Série A : Polynômes et Trigonométrie (20 exercices)}
\begin{multicols}{3}
    \setlength{\columnseprule}{0.9pt}

\begin{enumerate}
    \item $\int_{0}^{\pi} x \sin(x) \, dx$
    \item $\int_{0}^{\pi/2} x \cos(x) \, dx$
    \item $\int_{0}^{\pi} (2x+1) \sin(x) \, dx$
    \item $\int_{0}^{\pi/2} (3x-2) \cos(x) \, dx$
    \item $\int_{0}^{\pi} x \sin(2x) \, dx$
    \item $\int_{0}^{\pi/4} x \cos(2x) \, dx$
    \item $\int_{0}^{\pi} x^2 \sin(x) \, dx$ (double IPP)
    \item $\int_{0}^{\pi} x^2 \cos(x) \, dx$
    \item $\int_{0}^{\pi/2} x \sin(x+\pi/4) \, dx$
    \item $\int_{0}^{\pi} (x^2+x) \sin(x) \, dx$
    \item $\int_{0}^{\pi/2} 2x \cos(3x) \, dx$
    \item $\int_{0}^{\pi} (1-x) \sin(x) \, dx$
    \item $\int_{0}^{\pi/3} x \sec^2(x) \, dx$
    \item $\int_{0}^{\pi} (x-1) \cos(2x) \, dx$
    \item $\int_{0}^{\pi/2} x^2 \sin(2x) \, dx$
    \item $\int_{-\pi}^{\pi} x \cos(x) \, dx$
    \item $\int_{0}^{2\pi} x \sin(x) \, dx$
    \item $\int_{0}^{\pi/2} (4x+3) \cos(x) \, dx$
    \item $\int_{0}^{\pi} x \sin(x/2) \, dx$
    \item $\int_{0}^{\pi} (x^2-1) \cos(x) \, dx$
\end{enumerate}
\end{multicols}

\subsection*{Série B : Polynômes et Exponentielles (20 exercices)}
\begin{multicols}{3}
    \setlength{\columnseprule}{0.9pt}

\begin{enumerate}
    \item $\int_{0}^{1} x e^x \, dx$
    \item $\int_{0}^{1} x e^{-x} \, dx$
    \item $\int_{0}^{1} (2x) e^{-x} \, dx$
    \item $\int_{0}^{\ln 2} x e^{2x} \, dx$
    \item $\int_{-1}^{1} (x+1) e^x \, dx$
    \item $\int_{0}^{2} (3x-1) e^{x/2} \, dx$
    \item $\int_{0}^{1} x^2 e^x \, dx$ (double IPP)
    \item $\int_{0}^{1} (x^2-2x) e^x \, dx$
    \item $\int_{0}^{3} x e^{-x} \, dx$
    \item $\int_{0}^{1} (4x+2) e^{2x} \, dx$
    \item $\int_{-1}^{0} x e^{-x} \, dx$
    \item $\int_{0}^{1} x^2 e^{-x} \, dx$
    \item $\int_{0}^{2} (x+3) e^{x} \, dx$
    \item $\int_{0}^{\ln 3} 2x e^{x} \, dx$
    \item $\int_{0}^{1} (1-x) e^x \, dx$
    \item $\int_{0}^{1} x^3 e^x \, dx$
    \item $\int_{0}^{1} x e^{3x+1} \, dx$
    \item $\int_{0}^{2} (x-1) e^{-2x} \, dx$
    \item $\int_{1}^{2} x e^x \, dx$
    \item $\int_{0}^{1} (x^2+1) e^{-x} \, dx$
\end{enumerate}
\end{multicols}

\subsection*{Série C : Fonctions Logarithmes (20 exercices)}
\begin{multicols}{3}
    \setlength{\columnseprule}{0.9pt}

\begin{enumerate}
    \item $\int_{1}^{e} \ln(x) \, dx$
    \item $\int_{1}^{e} 3\ln(x) \, dx$
    \item $\int_{1}^{e} x \ln(x) \, dx$
    \item $\int_{1}^{2} x^2 \ln(x) \, dx$
    \item $\int_{1}^{e} \frac{\ln(x)}{x^2} \, dx$
    \item $\int_{1}^{e} \sqrt{x} \ln(x) \, dx$
    \item $\int_{1}^{e} (\ln x)^2 \, dx$
    \item $\int_{e}^{e^2} \ln(x) \, dx$
    \item $\int_{1}^{e} (x+1) \ln(x) \, dx$
    \item $\int_{1}^{e} \frac{1}{x^2} \ln(x) \, dx$
    \item $\int_{1}^{4} \ln(\sqrt{x}) \, dx$
    \item $\int_{2}^{e} \ln(x-1) \, dx$
    \item $\int_{1}^{e} x^3 \ln(x) \, dx$
    \item $\int_{1}^{2} (2x-1) \ln(x) \, dx$
    \item $\int_{1}^{e} \ln(2x) \, dx$
    \item $\int_{1}^{e} \frac{\ln(x)}{\sqrt{x}} \, dx$
    \item $\int_{1}^{e} x^n \ln(x) \, dx$
    \item $\int_{1}^{2} \ln(x^2) \, dx$
    \item $\int_{e}^{2e} \ln(x/e) \, dx$
    \item $\int_{1}^{e} (x^2+x+1) \ln(x) \, dx$
\end{enumerate}
\end{multicols}


\section*{Exercice 19 : Pratique intensive de l'IPP}
\textit{Calculer les intégrales suivantes en posant judicieusement $u(x)$ et $v'(x)$.}

\subsection*{Série D : Variations Trigonométriques}
\begin{multicols}{3}
    \setlength{\columnseprule}{0.9pt}

\begin{enumerate}
    \item $\int_{0}^{\pi} x \sin(3x) \, dx$
    \item $\int_{0}^{\pi/2} 2x \cos(4x) \, dx$
    \item $\int_{0}^{\pi} (x+2) \sin(x/2) \, dx$
    \item $\int_{0}^{\pi} (5-x) \cos(x) \, dx$
    \item $\int_{0}^{\pi/4} x \sin(2x+\pi/4) \, dx$
    \item $\int_{0}^{\pi/2} x^2 \sin(x) \, dx$
    \item $\int_{0}^{\pi} e^x \cos(x) \, dx$
    \item $\int_{0}^{\pi} e^x \sin(x) \, dx$
    \item $\int_{0}^{1} x \arctan(x) \, dx$
    \item $\int_{0}^{\pi/2} x^2 \cos(2x) \, dx$
    \item $\int_{0}^{\pi} (x^2+1) \sin(x) \, dx$
    \item $\int_{0}^{\pi/2} x \sin^2(x) \cos(x) \, dx$
    \item $\int_{0}^{\pi} x \cos^2(x) \sin(x) \, dx$
    \item $\int_{0}^{\pi/6} x \sin(3x) \, dx$
    \item $\int_{-\pi}^{\pi} (x+1) \cos(x) \, dx$
    \item $\int_{0}^{\pi/4} \frac{x}{\cos^2(x)} \, dx$
    \item $\int_{0}^{\pi} e^{2x} \sin(3x) \, dx$
    \item $\int_{0}^{\pi/2} (x^2-4) \sin(x) \, dx$
    \item $\int_{0}^{\pi} x \cos(x/3) \, dx$
    \item $\int_{0}^{1} \arcsin(x) \, dx$
\end{enumerate}
\end{multicols}

\subsection*{Série E : Exponentielles et Puissances (20 exercices)}
\begin{multicols}{3}
    \setlength{\columnseprule}{0.9pt}

\begin{enumerate}
    \item $\int_{0}^{1} x^2 e^{2x} \, dx$
    \item $\int_{0}^{2} x^2 e^{-x} \, dx$
    \item $\int_{0}^{1} (x+3) e^{-2x} \, dx$
    \item $\int_{0}^{\ln 2} (2x+1) e^x \, dx$
    \item $\int_{-2}^{0} (x+2) e^x \, dx$
    \item $\int_{0}^{1} x^3 e^{-x} \, dx$
    \item $\int_{0}^{1} (x^2+x+1) e^x \, dx$
    \item $\int_{0}^{1} x e^{ax+b} \, dx$
    \item $\int_{1}^{2} (x-1) e^{2x} \, dx$
    \item $\int_{0}^{0.5} x e^{-4x} \, dx$
    \item $\int_{0}^{1} x^2 e^{x/2} \, dx$
    \item $\int_{-1}^{1} |x| e^x \, dx$
    \item $\int_{0}^{1} (2x^2-1) e^{-x} \, dx$
    \item $\int_{0}^{2} x e^{x-2} \, dx$
    \item $\int_{0}^{1} \frac{x}{e^x} \, dx$
    \item $\int_{0}^{\ln 5} x e^x \, dx$
    \item $\int_{0}^{1} (x+1)^2 e^x \, dx$
    \item $\int_{0}^{1} x^2 e^{-3x} \, dx$
    \item $\int_{1}^{e} x e^{\ln x} \, dx$
    \item $\int_{0}^{1} (1-x^2) e^x \, dx$
\end{enumerate}
\end{multicols}

\subsection*{Série F : Logarithmes et Fonctions Composées }
\begin{multicols}{3}
    \setlength{\columnseprule}{0.9pt}

\begin{enumerate}
    \item $\int_{1}^{e} x^4 \ln(x) \, dx$
    \item $\int_{1}^{e} \frac{\ln(x)}{x^3} \, dx$
    \item $\int_{1}^{2} (x^2+1) \ln(x) \, dx$
    \item $\int_{1}^{e} \ln(x^2+x) \, dx$
    \item $\int_{1}^{e} \ln(1/x) \, dx$
    \item $\int_{e}^{e^2} \frac{\ln(x)}{\sqrt{x}} \, dx$
    \item $\int_{1}^{4} \sqrt{x} \ln(\sqrt{x}) \, dx$
    \item $\int_{1}^{e} x \ln(x^2) \, dx$
    \item $\int_{2}^{3} (x-2) \ln(x) \, dx$
    \item $\int_{1}^{e} \frac{\ln(x)+1}{x} \, dx$
    \item $\int_{1}^{e} (x^2-1) \ln(x) \, dx$
    \item $\int_{1}^{e} \ln(x+1) \, dx$
    \item $\int_{1}^{e} x \ln(2x) \, dx$
    \item $\int_{1}^{2} x^n \ln(x) \, dx$
    \item $\int_{1}^{e} (\ln x)^3 \, dx$

\end{enumerate}
\end{multicols}



\end{itshare}
\end{document}