add two first valid sequences
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// préambule asymptote
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usepackage("amsmath,amssymb");
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usepackage("inputenc","utf8");
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usepackage("icomma");
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import lib_jl;
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unitsize(1cm,2cm);
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xlimits(-1.25,10);
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ylimits(-0.5,3.5);
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xaxis(BottomTop,xmin=-1.25,xmax=10,Ticks("%",extend=true,Step=1),p=linewidth(0.5pt)+gray(0)+dotted);
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yaxis(LeftRight,ymin=-0.5,ymax=3.5,Ticks("%",extend=true,Step=1),p=linewidth(0.5pt)+gray(0)+dotted);
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xlimits(-1.25,10,Crop);
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ylimits(-0.5,3.5,Crop);
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real f(real x) {return 1/8*(x-5)^2+3/2;}
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pair O=(0,0), I=(1,0), J=(0,1), A=(1,1), D=(3,0), E=(8,0), F=(3,f(3)), G=(8,f(8));
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label("$O$",O,4.5000(-0.2121,-0.2121));
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label("$I$",I,1.5*S);
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label("$J$",J,1.5*W);
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label("$A$",A, .125*NE);
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filldraw(O--I--A--J--cycle, lightgray, linewidth(1pt));
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path Cf=graph(f, 2.5, 8.5, n=400);
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draw(Cf, linewidth(1pt));
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path domaineD = buildcycle(F--D--E--G, graph(f,3,8));
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fill(domaineD, pattern("hachures1"));
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//draw(F--G, linewidth(1.5pt));
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draw((D.x,D.y-.5)--(F.x,F.y+.5), gray(0)+dashed);
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draw((E.x,E.y-.5)--(G.x,G.y+.5), gray(0)+dashed);
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label("Domaine $D$", (5.5,1), fontsize(18pt), Fill(white));
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label("$\int_{a}^{b} f(t)\; dt$", (5.5,.5), fontsize(18pt), Fill(white));
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label("$a$", D, 1.75*S, fontsize(18pt), Fill(white));
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label("$b$", E, 1.25*S, fontsize(18pt), Fill(white));
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xaxis(axis=YEquals(0),xmin=-1.25,xmax=10,Ticks("%",NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true);
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yaxis(axis=XEquals(0),ymin=-0.5,ymax=3.5,Ticks("%",NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true);
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shipout(bbox(0.1cm,0.1cm,white));
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// préambule asymptote
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usepackage("amsmath,amssymb");
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usepackage("inputenc","utf8");
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usepackage("icomma");
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import lib_jl;
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unitsize(1cm,1cm);
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real xmin=-5.5, ymin=-1.5, xmax=5.5, ymax=5.5;
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xlimits(xmin,xmax);
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ylimits(ymin,ymax);
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xaxis(BottomTop,xmin=xmin,xmax=xmax,Ticks("%",extend=true,Step=1),p=linewidth(0.5pt)+gray(0)+dotted);
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yaxis(LeftRight,ymin=ymin,ymax=ymax,Ticks("%",extend=true,Step=1),p=linewidth(0.5pt)+gray(0)+dotted);
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real F(real x) {return .5*(x+5)*(.5x^2+.25x+.5);}
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draw(graph(F,-6,4,n=400),linewidth(1pt)+deepred+solid);
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xlimits(xmin,xmax,Crop);
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ylimits(ymin,ymax,Crop);
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xaxis(axis=YEquals(0),xmin=xmin,xmax=xmax,Ticks("%",beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true);
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yaxis(axis=XEquals(0),ymin=ymin,ymax=ymax,Ticks("%",beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true);
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pair O=(0,0), I=(1,0), J=(0,1), A=(-4,0), B=(1,0), C=(-1.5,0);
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labelx(Label("$O$",NoFill), 0, SW);
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label("$I$",I,SE);
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label("$J$",J,NW);
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draw(line(A-(0,2),A+(0,2)),deepblue+pen1);
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label("$a$",A,SE,deepblue);
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path integrale1=buildcycle((A.x, F(A.x))--A--C--(C.x, F(C.x)),graph(F,A.x,C.x));
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add("integrale1",hatch(H=.5mm,dir=NE,lightred));
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fill(integrale1,pattern("integrale1"));
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label(scale(.9)*"$\int_{a}^{c} f(x)\; dx$",((A.x+C.x)/2,2),deepred);
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path integrale2=buildcycle((C.x, F(C.x))--C--B--(B.x, F(B.x)),graph(F,C.x,B.x));
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add("integrale2",hatch(H=.5mm,dir=NW,lightgreen));
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fill(integrale2,pattern("integrale2"));
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label(scale(.9)*"$\int_{c}^{b} f(x)\; dx$",((C.x+B.x)/2,.5),deepgreen);
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draw(line(B-(0,2),B+(0,2)),deepblue+pen1);
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label("$b$",B,SW,deepblue);
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draw(line(C-(0,2),C+(0,2)),deepgreen+pen1);
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label("$c$",C,SW,deepgreen);
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shipout(bbox(0.1cm,0.1cm,white));
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../images/
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% !TeX root = prof_-_calcul_integral_-_2023_1.tex
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\partie{Primitives}
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\noindent
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Soit les fonctions $f(x) = 2x^2 + 3x + 5$ et $F(x) = \dfrac23 x^3 + \dfrac32 x^2 + 5x + 3$, définies sur $ \mathbb{R} $.\\
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\noindent
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Quel lien existe entre ces deux fonctions?
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\vspace{2cm}
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\hrule
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\sspartie{Définitions}
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\noindent
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Dans cette partie, on considère une fonction $f$ définie sur un intervalle $I$ de $ \mathbb{R} $.
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\vspace{.4cm}
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% Définition
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\begin{definition}[primitive]
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On dit que $ F $ est une \textbf{primitive} de $ f $ sur $ I $ lorsque $ F $ est dérivable sur $ I $ et $ F'=f $.
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\end{definition}
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\begin{propriete}[]
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Toute fonction continue sur un intervalle $I$ admet une primitive sur cet intervalle.
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\end{propriete}
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\begin{application}
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\vspace{-.4cm}
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$ f $ est la fonction définie sur $ \R $ par $ f(x) = e^x+2 $.\\
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Déterminer une primitive de la fonction $f$ sur $\R$.
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\vspace{1cm}
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\end{application}
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\vspace{.4cm}
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\begin{propriete}[primitives d'une même fonction]
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Deux primitives d'une même fonction continue sur un intervalle diffèrent d'une constante.
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\Cad que si $F$ est une primitive de $f$ sur $I$, alors la fonction $G_k$ définie sur $I$ par $G_k(x) = F(x) + k$ avec $k \in \mathbb{R}$ est aussi une primitive de $f$.
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\end{propriete}
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\vspace{.4cm}
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\begin{propriete}
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Soit $x_0$ et $y_0$ deux réels de $I$.\\
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Il existe une unique primitive $F$ de $f$ telle que $F(x_0) = y_0$.\\
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Cette valeur s'appelle une condition initiale.
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\end{propriete}
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\begin{application}[]
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Soit la fonction $f(x) = 5x^3 - 8x^2 + \dfrac54$ définie sur $ R $.
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\begin{questions}
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\item Déterminer l'ensemble des primitives $F_k$, où $k \in \R$ de la fonction $f$.
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\item Déterminer la primitive de la fonction $f$ prenant la valeur $25$ en $3$.
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\end{questions}
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\end{application}
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\sspartie{Primitives des fonctions usuelles}
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\begin{center}
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\small
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\begin{tabularx}{1\textwidth}{| >{\centering\arraybackslash}X | >{\centering\arraybackslash}X | >{\centering\arraybackslash}X |}
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\hline
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\textbf{Fonction $\boldsymbol{f}$} & \textbf{Une primitive $\boldsymbol{F}$} & \textbf{Sur l'intervalle $\boldsymbol{I}$}\\
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\hline
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\rule[-1ex]{0pt}{5ex}$ f(x)=a $, $ a \in \R $ & $ F(x)=ax+C $ & $ \R $\\[2ex]
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\hline
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\rule[-1ex]{0pt}{6ex}$ f(x)=x^n $, $ n \in \Z $ et $ n \neq -1 $ & $ F(x)=\dfrac{x^{n+1}}{n+1}+C $ & $\begin{array}{ccr}\R&\text{ si }n\in\N\\ ]-\infty\ ;\ 0[ \text{ ou } ]0\ ;\ +\infty[&\text{ si }n<0\\\end{array}$\\[3ex]
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\hline
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\rule[-1ex]{0pt}{6ex}$ f(x)=\dfrac1{x^2} $ & $ F(x)=-\dfrac1{x} +C $ & $ ]-\infty\ ;\ 0[ $ ou $ ]0\ ;\ +\infty[ $\\[2ex]
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\hline
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\rule[-1ex]{0pt}{6ex}$ f(x)=\dfrac1{x} $ & $ F(x)=\ln x +C $ & $ ]0\ ;\ +\infty[ $\\[2ex]
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\hline
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\rule[-1ex]{0pt}{6ex}$ f(x)=\dfrac1{\sqrt{x}} $ & $ F(x)=2\sqrt{x} +C $ & $ ]0\ ;\ +\infty[ $\\[4ex]
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\hline
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\rule[-1ex]{0pt}{5ex}$ f(x)=e^{x} $ & $ F(x)=e^x +C$ & $ \R $\\[2ex]
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\hline
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\rule[-1ex]{0pt}{5ex}$ f(x)=\cos{x} $ & $ F(x)=\sin{x} +C$ & $ \R $\\[2ex]
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\hline
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\rule[-1ex]{0pt}{5ex}$ f(x)=\sin{x} $ & $ F(x)=-\cos{x} +C$ & $ \R $\\[2ex]
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\hline
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\end{tabularx}
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\end{center}
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\vspace{.4cm}
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\begin{application}[calcul de primitives (1)]
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Calculer les primitives des fonctions suivantes:
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\begin{tasks}(4)[style=enumerate]
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\task $f(x) = x$
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\task $f(x) = x^3$
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\task $f(x) = 3x^5+1$
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\task $f(x) = \sqrt{x}$
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\task $f(x) = \dfrac1{x^2}$
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\task $f(x) = \dfrac1{\sqrt{x}}$
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\task $f(x) = \dfrac{-2}{x^4} + \dfrac1{\sqrt{x}}$
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\task $f(x) = 7\cos x - 2\sin x$
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\end{tasks}
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\end{application}
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\sspartie{Primitives et composition}
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\begin{center}
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\small
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\begin{tabularx}{1\textwidth}{| >{\centering\arraybackslash}X | >{\centering\arraybackslash}X | >{\centering\arraybackslash}X |}
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\hline
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\textbf{Fonction $\boldsymbol{f}$} & \textbf{Une primitive $\boldsymbol{F}$ sur $\boldsymbol{I}$} & \textbf{Condition sur $\boldsymbol{u}$}\\
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\hline
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\rule[-1ex]{0pt}{6ex}$ a\, u' $, $ a \in \R $ & $ a\, u + C $ & \\[3ex]
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\hline
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\rule[-1ex]{0pt}{6ex}$ u'+v' $ & $ u + v + C $ & \\[3ex]
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\hline
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\rule[-1ex]{0pt}{6ex}$ u'\, u^n $, $ n \in \Z $ et $ n \notin \{-1\; ;\ 0\} $ & $ \dfrac{u^{n+1}}{n+1}+C $ & $u(x) \neq 0 \text{ pour } x\in I \text{ si } n\le -2$\\[3ex]
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\hline
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\rule[-1ex]{0pt}{6ex}$ \dfrac{u'}{u} $ & $ \ln(u) +C $ & $ u(x)>0,\ x \in I $\\[2ex]
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\hline
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\rule[-1ex]{0pt}{6ex}$ \dfrac{u'}{u^2} $ & $ -\dfrac1{u} +C $ & $ u(x)\neq 0,\ x \in I $\\[2ex]
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\hline
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\rule[-1ex]{0pt}{6ex}$ \dfrac{u'}{\sqrt{u}} $ & $ 2\sqrt{u} +C $ & $ u(x)>0,\ x \in I $\\[2ex]
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\hline
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\rule[-1ex]{0pt}{6ex}$ u'e^{u} $ & $ e^{u} +C $ & \\[2ex]
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\hline
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\rule[-1ex]{0pt}{6ex}$ \cos(\omega\, x + \varphi) $, $ \omega \in \R $ et $ \varphi \in \R $ & $ \dfrac1{\omega}\sin(\omega\, x + \varphi) + C $ & \\[3ex]
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\hline
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\rule[-1ex]{0pt}{6ex}$ \sin(\omega\, x + \varphi) $, $ \omega \in \R $ et $ \varphi \in \R $ & $ -\dfrac1{\omega}\cos(\omega\, x + \varphi) + C $ & \\[3ex]
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\hline
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\end{tabularx}
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\end{center}
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\vspace{.4cm}
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\begin{application}[calcul de primitives (2)]
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Calculer les primitives des fonctions suivantes:
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\begin{tasks}(4)[style=enumerate]
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\task $ f(x) = \dfrac{2x}{x^2+1} $
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\task $ f(x) = (x+1)\e^{x^2+2x+1} $
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\task $ f(x) = \dfrac{x}{\sqrt{x^2+1}} $
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\task $ f(x) = \cos(2x + 7) $
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\task $ f(x) = \dfrac{x^3}{\left( x^4+1 \right)} $
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\task $ f(x) = \sin x\, \cos^3 x $
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\task $ f(x) = \tan x $
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\task $ f(x) = \dfrac{\sin x}{\cos^2 x } $
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\end{tasks}
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\end{application}
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% !TeX root = prof_-_calcul_integral_-_2023_1.tex
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\newpage
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\partie{Intégration}
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\sspartie{Notion d'aire}
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\begin{definition}[intégrale]
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On appelle intégrale de la fonction $f$ entre $a$ et $b$ est l'aire signée sous la courbe de $f$.
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\begin{center}
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\includegraphics[scale=.3]{fig1.png}
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\end{center}
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Ceci signifie qu'on compte positivement les aires situées au-dessus de l'axe des abscisses et négativement celles situées en-dessous, puis qu'on en fait la somme.\\
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\noindent
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On note alors cette intégrale \(\int_{a}^{b} f(x)\; dx\) et on lira <<intégrale de \(a\) à \(b\) de \(f(x)\)>>.
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\end{definition}
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\noindent
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$\int_{a}^{b} f(x)\; dx$ peut être interpréter comme la <<somme infinie>> des aires $f(x) \times dx$ des rectangles infinitésimaux de hauteur $f(x)$ et de largeur $dx$.
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\vspace{-8pt}
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\begin{center}
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\begin{tabular}[c]{ccc}
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\includegraphics[scale=.25]{fig2.png} & \hspace{.4cm} & \includegraphics[scale=.2]{fig3.png}
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\end{tabular}
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\end{center}
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\begin{multicols}{2}
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\noindent
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Sur l'intervalle $[a\ ;\ b]$ l'aire sous la courbe est la surface du domaine hachuré $D$.\\
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Cette aire est donc obtenue par calcul de \(\int_{a}^{b} f(t)\; dt\).
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\noindent
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L'unité d'aire est donnée par la surface du rectangle $OIAJ$.\\\\
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\columnbreak
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\begin{center}
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\includegraphics[scale=.6]{fig4.pdf}
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\end{center}
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\end{multicols}
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\sspartie{Calcul d'intégrales}
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\begin{propriete}[calcul d'une intégrale]
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Soit $F$ une primitive de $f$ sur l'intervalle $I$. Alors: \qquad $\int_{a}^{b} f(x)\; dx = [F(x)]_{a}^{b} = F(b)-F(a)$.
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\end{propriete}
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\begin{remarque}
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Dans l'écriture \(\int_{a}^{b} f(x)\; dx\), la lettre $x$ est une variable <<muette>>. Ainsi: $\int_{a}^{b} f(x)\; dx = \int_{a}^{b} f(y)\; dy = \int_{a}^{b} f(t)\; dt = \int_{a}^{b} f$
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\end{remarque}
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\medskip
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\begin{propriete}[]
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\begin{tasks}(3)[style=enumerate]
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\task \(\int_{a}^{a} f(x)\; dx = 0 \)
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\task \(\int_{a}^{b} f(x)\; dx = -\int_{b}^{a} f(x)\; dx \)
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\task \(\int_{a}^{a} k\; dx = k\, \left(b-a\right) \)
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\end{tasks}
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\end{propriete}
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\medskip
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\begin{application}[calculer une intégrale depuis une primitive]
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Calculer :
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% Questions
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\begin{tasks}(2)
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\task $ A = \int_{1}^{4} \dfrac3{x^2}\; dx $
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\task $ B = \int_{2}^{5} (3t^2+4t-5)\; dt $
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\task $ C = \int_{-1}^{1} 2t^2-1\; dt $
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\task $ D = \int_{-1}^{1} 2+e^{-2x}\; dx $
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\end{tasks}
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\end{application}
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\sspartie{Propriétés de l'intégrale}
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\begin{propriete}[relation de Chasles]
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Soit \(f\) une fonction continue sur un intervalle $ I=[a\ ;\ b] $. Soit $ c $ un réel de $ I $.
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\begin{center}
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\begin{tabular}{cc}
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\rule[-1ex]{0pt}{2.5ex}
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\begin{minipage}{.4\textwidth}
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\[
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\int_{a}^{b} f(x)\; dx = \int_{a}^{c} f(x)\; dx + \int_{c}^{b} f(x)\; dx
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\]
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\end{minipage}
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&
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\begin{minipage}{.7\textwidth}
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\includegraphics[scale=.7]{fig5.pdf}
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\end{minipage}
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\end{tabular}
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||||
\end{center}
|
||||
\end{propriete}
|
||||
|
||||
\medskip
|
||||
|
||||
\begin{propriete}[linéarité]
|
||||
\vspace{-.4cm}
|
||||
Soient \(f\) et \(g\) deux fonctions continues sur un intervalle $ I=[a\ ;\ b] $ et $\lambda$ un réel quelconque.
|
||||
\[
|
||||
\int_{a}^{b} (f(x) + \lambda\, g(x))\; dx = \int_{a}^{b} f(x)\; dx + \lambda\, \int_{a}^{b} g(x)\; dx
|
||||
\]
|
||||
\end{propriete}
|
||||
|
||||
\medskip
|
||||
|
||||
\begin{propriete}[inégalité]
|
||||
Soient \(f\) et \(g\) deux fonctions continues sur un intervalle $ I=[a\ ;\ b] $.
|
||||
\begin{itemize}
|
||||
\item Si pour tout $ x \in [a\ ;\ b], f(x) \geq 0 $, alors $ \int_{a}^{b} f(x)\; dx \geq 0 $ ;
|
||||
\item Si pour tout $ x \in [a\ ;\ b], f(x) \geq g(x) $, alors $ \int_{a}^{b} f(x)\; dx \geq \int_{a}^{b} g(x)\; dx $.
|
||||
\end{itemize}
|
||||
\end{propriete}
|
||||
|
||||
\medskip
|
||||
|
||||
\begin{application}[]
|
||||
\vspace{-.4cm}
|
||||
\small
|
||||
% Questions
|
||||
\begin{questions}
|
||||
\item Calculer $ \int_{1}^{e} \dfrac1{x}\; dx $.
|
||||
\item
|
||||
% Questions
|
||||
\begin{questions}
|
||||
\item Justifier que la fonction $ x \mapsto \dfrac1{x+1} $ a pour primitive sur $ ]-1\ ;\ +\infty[ $ la fonction $ x \mapsto \ln(x+1) $.\\
|
||||
On admettra que la dérivée d'une fonction sous la forme $ \ln(u) $ est $ \dfrac{u'}{u} $.
|
||||
\item Calculer $ \int_{1}^{e} \dfrac1{1+x}\; dx $.
|
||||
\end{questions}
|
||||
\item Démontrer que pour tout $ x > 0 $, $ \dfrac1{x(x+1)} = \dfrac1{x}-\dfrac1{x+1}$
|
||||
\item En déduire la valeur de $ \int_{1}^{e} \dfrac1{x(x+1)}\; dx $.
|
||||
\end{questions}
|
||||
\end{application}
|
||||
|
||||
\medskip
|
||||
|
||||
\begin{propriete}[valeur moyenne d'une fonction sur un intervalle]
|
||||
% \underline{\textbf{Graphiquement :}}
|
||||
\begin{center}
|
||||
\begin{tabular}{l c r}
|
||||
\begin{minipage}{.4\textwidth}
|
||||
\noindent
|
||||
On appelle valeur moyenne de $ f $ sur $ I $ le nombre réel:
|
||||
\[
|
||||
m = \dfrac1{b-a}\, \int_{a}^{b} f(x)\, dx
|
||||
\]\\
|
||||
|
||||
\noindent
|
||||
L'aire sous la courbe représentative de $ f $ (en rouge ci-contre) est égale à l'aire sous la droite d'équation $ y = m $ (en bleu).
|
||||
\end{minipage}
|
||||
&
|
||||
\hspace{.5cm}
|
||||
&
|
||||
\begin{minipage}{.7\textwidth}
|
||||
\includegraphics[scale=.5]{fig6.pdf}
|
||||
\end{minipage}
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\end{propriete}
|
||||
|
||||
\medbreak
|
||||
@@ -0,0 +1,47 @@
|
||||
% !TeX root = prof_-_calcul_integral_-_2023_1.tex
|
||||
|
||||
\partie{Intégration par parties}
|
||||
|
||||
\begin{propriete}[]
|
||||
\vspace{-.4cm}
|
||||
Soit $ u $ et $ v $ deux fonctions dérivables sur $ [a\ ;\ b] $ de dérivées $ u' $ et $ v' $ continues, alors :
|
||||
|
||||
\[
|
||||
\int_{a}^{b} uv'(x)\; dx = \left[ u\; v(x)\right ]_{a}^{b} - \int_{a}^{b} u'v(x)\; dx
|
||||
\]
|
||||
\end{propriete}
|
||||
|
||||
% remarque
|
||||
\begin{remarque}
|
||||
Le choix de $ u $ et $ v' $ est à faire judicieusement puisqu'il faut pouvoir trouver une primitive de $ u'v $.
|
||||
\end{remarque}
|
||||
|
||||
\medskip
|
||||
|
||||
\begin{application}[intégration par partie]
|
||||
\vspace{-.4cm}
|
||||
Calculer $ \int_{0}^{1} x\e^{x}\; dx $.
|
||||
%
|
||||
% \noindent
|
||||
% On ne peut pas trouver une primitive de $ x \mapsto x\e^{x} $ car non de la forme $ u'\e^{u} $.\\
|
||||
% On intègre par partie :
|
||||
%
|
||||
% \[
|
||||
% \begin{array}{rcl}
|
||||
% u(x) = x & & u'(x) = 1 \\
|
||||
% v'(x) = \e^{x} & & v(x) = \e^{x}
|
||||
% \end{array}
|
||||
% \]
|
||||
%
|
||||
% \[
|
||||
% \int_{0}^{1} x\e^{x}\; dx = \left[ x\e^{x} \right]_{0}^{1} - \int_{0}^{1} \e^{x} \times 1\; dx = \left[ x\e^{x} \right]_{0}^{1} - \int_{0}^{1} \e^{x} \; dx = \left[ x\e^{x} \right]_{0}^{1} - \left[\e^{x} \right]_{0}^{1} = (\e - 0) - (\e - 1) = 1.
|
||||
% \]
|
||||
\end{application}
|
||||
|
||||
\medskip
|
||||
|
||||
% application
|
||||
\begin{application}[déterminer une primitive]
|
||||
\vspace{-.4cm}
|
||||
Déterminer une primitive de $ \ln $ sur $ ]0\ ;\ +\infty[ $.
|
||||
\end{application}
|
||||
@@ -0,0 +1,623 @@
|
||||
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|
||||
! Undefined control sequence.
|
||||
l.27 \reset@newl@bel
|
||||
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
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|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.28 \xpglanginauxtrue
|
||||
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
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||||
! Not loaded.
|
||||
\selectlanguage ...ing it}\errmessage {Not loaded}
|
||||
|
||||
l.29 \selectlanguage
|
||||
*{french}
|
||||
Selecting a language requires a package supporting it
|
||||
|
||||
|
||||
! LaTeX Error: Missing \begin{document}.
|
||||
|
||||
See the LaTeX manual or LaTeX Companion for explanation.
|
||||
Type H <return> for immediate help.
|
||||
...
|
||||
|
||||
l.29 \selectlanguage *
|
||||
{french}
|
||||
You're in trouble here. Try typing <return> to proceed.
|
||||
If that doesn't work, type X <return> to quit.
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||||
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||||
! Undefined control sequence.
|
||||
l.31 \xpglanginauxfalse
|
||||
|
||||
The control sequence at the end of the top line
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||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
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||||
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||||
! Undefined control sequence.
|
||||
l.34 \savepicturepage
|
||||
{pgfid1}{1}
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||||
The control sequence at the end of the top line
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||||
of your error message was never \def'ed. If you have
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||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
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||||
|
||||
! Undefined control sequence.
|
||||
l.35 \pgfsyspdfmark
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||||
{pgfid1}{1305270}{27538278}
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||||
The control sequence at the end of the top line
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||||
of your error message was never \def'ed. If you have
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||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.36 \savepicturepage
|
||||
{pgfid2}{1}
|
||||
The control sequence at the end of the top line
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||||
of your error message was never \def'ed. If you have
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||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.37 \pgfsyspdfmark
|
||||
{pgfid2}{1305270}{23310906}
|
||||
The control sequence at the end of the top line
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||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.38 \savepicturepage
|
||||
{pgfid3}{1}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.39 \pgfsyspdfmark
|
||||
{pgfid3}{1305270}{17174441}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.40 \savepicturepage
|
||||
{pgfid4}{1}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.41 \pgfsyspdfmark
|
||||
{pgfid4}{1305270}{11295568}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.42 \savepicturepage
|
||||
{pgfid5}{1}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.43 \pgfsyspdfmark
|
||||
{pgfid5}{1305270}{4539756}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.46 \savepicturepage
|
||||
{pgfid6}{2}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.47 \pgfsyspdfmark
|
||||
{pgfid6}{1305270}{44980978}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.49 \savepicturepage
|
||||
{pgfid7}{2}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.50 \pgfsyspdfmark
|
||||
{pgfid7}{1305270}{14660377}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.52 \savepicturepage
|
||||
{pgfid8}{3}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.53 \pgfsyspdfmark
|
||||
{pgfid8}{1305270}{18097414}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.58 \savepicturepage
|
||||
{pgfid9}{4}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.59 \pgfsyspdfmark
|
||||
{pgfid9}{1305270}{33508000}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.60 \savepicturepage
|
||||
{pgfid10}{4}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.61 \pgfsyspdfmark
|
||||
{pgfid10}{1436342}{17297253}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.63 \pagesLTS@ifcounter
|
||||
{pagesLTS.arabic.1.local.cnt}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
|
||||
! LaTeX Error: No counter 'pagesLTS.arabic.1.local.cnt' defined.
|
||||
|
||||
See the LaTeX manual or LaTeX Companion for explanation.
|
||||
Type H <return> for immediate help.
|
||||
...
|
||||
|
||||
l.64 \setcounter{pagesLTS.arabic.1.local.cnt}{4}
|
||||
|
||||
Your command was ignored.
|
||||
Type I <command> <return> to replace it with another command,
|
||||
or <return> to continue without it.
|
||||
|
||||
|
||||
! LaTeX Error: No counter 'pagesLTS.pagenr' defined.
|
||||
|
||||
See the LaTeX manual or LaTeX Companion for explanation.
|
||||
Type H <return> for immediate help.
|
||||
...
|
||||
|
||||
l.66 \setcounter{pagesLTS.pagenr}{4}
|
||||
|
||||
Your command was ignored.
|
||||
Type I <command> <return> to replace it with another command,
|
||||
or <return> to continue without it.
|
||||
|
||||
)
|
||||
\openout1 = `prof_-_calcul_integral_-_2023_1.aux'.
|
||||
|
||||
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 0.
|
||||
LaTeX Font Info: ... okay on input line 0.
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||||
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 0.
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||||
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||||
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||||
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||||
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||||
LaTeX Font Info: ... okay on input line 0.
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||||
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 0.
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||||
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||||
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 0.
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||||
LaTeX Font Info: ... okay on input line 0.
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||||
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 0.
|
||||
LaTeX Font Info: ... okay on input line 0.
|
||||
(/usr/share/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
|
||||
Overfull \hbox (42.78717pt too wide) in paragraph at lines 29--13
|
||||
\OT1/cmr/m/n/12 fid513052704539756 pg-fid62 pg-fid6130527044980978 pg-fid72 pg-fid7130527014660377
|
||||
[]
|
||||
|
||||
[Loading MPS to PDF converter (version 2006.09.02).]
|
||||
\scratchcounter=\count272
|
||||
\scratchdimen=\dimen153
|
||||
\scratchbox=\box55
|
||||
\nofMPsegments=\count273
|
||||
\nofMParguments=\count274
|
||||
\everyMPshowfont=\toks24
|
||||
\MPscratchCnt=\count275
|
||||
\MPscratchDim=\dimen154
|
||||
\MPnumerator=\count276
|
||||
\makeMPintoPDFobject=\count277
|
||||
\everyMPtoPDFconversion=\toks25
|
||||
) (/usr/share/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
|
||||
Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf
|
||||
Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 485.
|
||||
(/usr/share/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
|
||||
File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live
|
||||
))
|
||||
*\makeatletter%
|
||||
|
||||
*\let\ASYencoding\f@encoding%
|
||||
|
||||
*\let\ASYfamily\f@family%
|
||||
|
||||
*\let\ASYseries\f@series%
|
||||
|
||||
*\let\ASYshape\f@shape%
|
||||
|
||||
*\makeatother%
|
||||
|
||||
*\fontsize{12}{14.4}\selectfont%
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}%
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\setbox\ASYbox=\hbox{}
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\immediate\write16{>dim(\the\wd\ASYbox)dim}
|
||||
>dim(0.0pt)dim
|
||||
|
||||
*\immediate\write16{>dim(\the\ht\ASYbox)dim}
|
||||
>dim(0.0pt)dim
|
||||
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||||
*\immediate\write16{>dim(\the\dp\ASYbox)dim}
|
||||
>dim(0.0pt)dim
|
||||
|
||||
*\fontsize{12}{14.4}\selectfont%
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}%
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\setbox\ASYbox=\hbox{$O$}
|
||||
LaTeX Font Info: Trying to load font information for U+esvect on input line 0.
|
||||
(/usr/share/texmf-dist/tex/latex/esvect/uesvect.fd
|
||||
File: uesvect.fd
|
||||
)
|
||||
LaTeX Font Info: Trying to load font information for U+rsfs on input line 0.
|
||||
(/usr/share/texmf-dist/tex/latex/jknapltx/ursfs.fd
|
||||
File: ursfs.fd 1998/03/24 rsfs font definition file (jk)
|
||||
)
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\immediate\write16{>dim(\the\wd\ASYbox)dim}
|
||||
>dim(9.30602pt)dim
|
||||
|
||||
*\immediate\write16{>dim(\the\ht\ASYbox)dim}
|
||||
>dim(8.2pt)dim
|
||||
|
||||
*\immediate\write16{>dim(\the\dp\ASYbox)dim}
|
||||
>dim(0.0pt)dim
|
||||
|
||||
*\fontsize{12}{14.4}\selectfont%
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}%
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\setbox\ASYbox=\hbox{$I$}
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\immediate\write16{>dim(\the\wd\ASYbox)dim}
|
||||
>dim(6.12566pt)dim
|
||||
|
||||
*\immediate\write16{>dim(\the\ht\ASYbox)dim}
|
||||
>dim(8.2pt)dim
|
||||
|
||||
*\immediate\write16{>dim(\the\dp\ASYbox)dim}
|
||||
>dim(0.0pt)dim
|
||||
|
||||
*\fontsize{12}{14.4}\selectfont%
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}%
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\setbox\ASYbox=\hbox{$J$}
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\immediate\write16{>dim(\the\wd\ASYbox)dim}
|
||||
>dim(7.60553pt)dim
|
||||
|
||||
*\immediate\write16{>dim(\the\ht\ASYbox)dim}
|
||||
>dim(8.2pt)dim
|
||||
|
||||
*\immediate\write16{>dim(\the\dp\ASYbox)dim
|
||||
Binary file not shown.
@@ -0,0 +1,29 @@
|
||||
% File : prof_-_calcul_integral_-_2023_1.tex
|
||||
% Author : Jeff Lance <email@jefflance.me>
|
||||
% Date : 18.08.2023 12:20:01
|
||||
% Last Modified Date: 18.08.2023 12:20:01
|
||||
% Last Modified By : Jeff Lance <email@jefflance.me>
|
||||
|
||||
\documentclass[11pt]{jl-cours}
|
||||
\usepackage{cancel}
|
||||
\usepackage{asymptote}
|
||||
|
||||
\begin{document}
|
||||
\pagenumbering{arabic}
|
||||
|
||||
\titre{\Jd\ Calcul intégral}{\Jd\ Calcul intégral}
|
||||
\lohead*{BTS2E}
|
||||
\rofoot*{\anneescolaire}
|
||||
|
||||
Dans ce document, $ f $ est une fonction définie et continue sur un intervalle $ I=[a\ ;\ b] $ de $ \R $.
|
||||
|
||||
% Primitives
|
||||
\import{.}{partieI.tex}
|
||||
|
||||
% Intégration
|
||||
\import{.}{partieII.tex}
|
||||
|
||||
% Intégration par parties
|
||||
\import{.}{partieIII.tex}
|
||||
|
||||
\end{document}
|
||||
Reference in New Issue
Block a user