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analyse/main.tex
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analyse/main.tex
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\tableofcontents
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\clearpage
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\section{Coin par c\oe{}ur}
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\paragraph{Trigonométrie}
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\begin{tabular}{c|ccccc}
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\toprule
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x & 0 & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{\pi}{3}$ & $\frac{\pi}{2}$ \\
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\midrule
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$\sin{x}$ & 0 & $\frac{1}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{3}}{2}$ & 1 \\
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\midrule
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$\cos{x}$ & 1 & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{1}{2}$ & 0 \\
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\midrule
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$\frac{\sin{x}}{\cos{x}} = \tan{x}$ & 0 & $\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ & 1 & $\sqrt{3}$ & impossible \\
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\bottomrule
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\end{tabular}
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\paragraph{Trigonométrie}
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\begin{tabular}{c|ccccc}
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\toprule
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x & 0 & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{\pi}{3}$ & $\frac{\pi}{2}$ \\
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\midrule
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$\sin{x}$ & 0 & $\frac{1}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{3}}{2}$ & 1 \\
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\midrule
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$\cos{x}$ & 1 & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{1}{2}$ & 0 \\
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\midrule
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$\frac{\sin{x}}{\cos{x}} = \tan{x}$ & 0 & $\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ & 1 & $\sqrt{3}$ & impossible \\
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\bottomrule
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\end{tabular}
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\paragraph{Exponentielle et Logarithme}
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\paragraph{Exponentielle et Logarithme}
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\hfill
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$e^0 = 1 ; e^1 = e$
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\hfill
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$\ln{0} = \text{impossible~; } \ln{1} = 0 \text{~; } \ln{e} = 1$
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\hfill
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$e^0 = 1 ; e^1 = e$
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\hfill
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$\ln{0} = \text{impossible~; } \ln{1} = 0 \text{~; } \ln{e} = 1$
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\paragraph{Dérivées et Primitives}
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\paragraph{Dérivées et Primitives}
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\begin{multicols}{2}
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\begin{tabularx}{\linewidth}{YY}
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\toprule
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Primitive --- $f(x)$ & Dérivée --- $f'(x)$ \\
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\toprule
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\textcolor{red}{$a$} & 0 \\
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\midrule
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\textcolor{red}{$ax$} & \textcolor{red}{$a$} \\
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\midrule
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$\frac{1}{2} x^2$ & \textcolor{red}{$x$} \\
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\midrule
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\textcolor{red}{$x^n$} & \textcolor{red}{$nx^{n-1}$} \\
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\midrule
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\textcolor{red}{$\sqrt{x}$} & $\frac{1}{2\sqrt{x}}$ \\
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\midrule
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$\frac{2}{3} x\sqrt{x}$ & \textcolor{red}{$\sqrt{x}$} \\
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\midrule
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\textcolor{red}{$e^{ax}$} & \textcolor{red}{$ae^{ax}$} \\
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\midrule
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\textcolor{red}{$a^x$} & $a^x \ln{a}$ \\
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\midrule
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\textcolor{red}{$\ln{|x|}$} & \textcolor{red}{$\frac{1}{x}$} \\
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\midrule
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\textcolor{red}{$-\frac{1}{x}$} & \textcolor{red}{$\frac{1}{x^2}$} \\
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\midrule
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\textcolor{red}{$\cos{x}$} & \textcolor{red}{$-\sin{x}$} \\
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\midrule
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\textcolor{red}{$\sin{x}$} & \textcolor{red}{$\cos{x}$} \\
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\midrule
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\textcolor{red}{$\tan{x}$} & $1 + \tan^2{x} = \frac{1}{\cos^2{x}}$ \\
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\midrule
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$\cot{x}$ & $-1 - \cot^2{x} = \frac{-1}{\sin^2{x}}$ \\
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\midrule
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$\arccos{x}$ & $\frac{-1}{\sqrt{1 - x^2}}$ \\
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\midrule
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$\arcsin{x}$ & $\frac{1}{\sqrt{1 - x^2}}$ \\
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\midrule
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\textcolor{red}{$\arctan{x}$} & \textcolor{red}{$\frac{1}{1 + x^2}$} \\
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\bottomrule
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\end{tabularx}
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\columnbreak
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\begin{tabularx}{\linewidth}{lY}
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\toprule
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\multirow{2}{*}{Linéarité} & $(u + v)' = u' + v'$ \\
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& $(au)' = au'$ \\
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\midrule
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Produit & $(uv)' = u'v + uv'$ \\
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\midrule
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Inverse & $\left(\frac{1}{v}\right)' = - \frac{v'}{v^2}$ \\
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\midrule
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Quotient & $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ \\
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\midrule
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Composée & $(f(u))' = u'f'(u)$ \\
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\bottomrule
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\end{tabularx}
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\begin{tabularx}{\linewidth}{YY}
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\toprule
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Fonction & Primitive \\
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\toprule
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$u'u^n$ & $\frac{u^{n+1}}{n+1}$ \\
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\midrule
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$\frac{u'}{u^2}$ & $-\frac{1}{u}$ \\
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\midrule
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$\frac{u'}{\sqrt{u}}$ & $2\sqrt{u}$ \\
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\midrule
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$u'\cos{u}$ & $\sin{u}$ \\
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\midrule
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$u'\sin{u}$ & $-\cos{u}$ \\
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\midrule
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$\frac{u'}{u}$ & $\ln{|u|}$ \\
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\midrule
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$u'e^u$ & $e^u$ \\
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\midrule
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$\frac{u'}{1 + u^2}$ & $\arctan{u}$ \\
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\bottomrule
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\end{tabularx}
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\end{multicols}
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\paragraph{Intégrales}
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$\int_a^b f(x)\dif x = [F(x)]_a^b = F(b) - F(a)$
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\begin{multicols}{2}
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\begin{tabularx}{\linewidth}{YY}
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\toprule
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Intégration par parties~: & Intégration par changement de variables~: \\
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Primitive --- $f(x)$ & Dérivée --- $f'(x)$ \\
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\toprule
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$a$ & 0 \\
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\midrule
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$\int_a^b uv'\dif x = [uv]_a^b - \int_a^b u'v\dif x$ & $\int_a^b f(x)\dif x = \int_{u(a)}^{u(b)} f(u)\frac{\dif u}{u'}$ \\
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$ax$ & $a$ \\
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\midrule
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$\frac{1}{2} x^2$ & $x$ \\
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\midrule
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$x^n$ & $nx^{n-1}$ \\
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\midrule
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$\sqrt{x}$ & $\frac{1}{2\sqrt{x}}$ \\
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\midrule
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$\frac{2}{3} x\sqrt{x}$ & $\sqrt{x}$ \\
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\midrule
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$e^{ax}$ & $ae^{ax}$ \\
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\midrule
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$a^x$ & $a^x \ln{a}$ \\
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\midrule
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$\ln{|x|}$ & $\frac{1}{x}$ \\
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\midrule
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$-\frac{1}{x}$ & $\frac{1}{x^2}$ \\
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\midrule
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$\cos{x}$ & $-\sin{x}$ \\
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\midrule
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$\sin{x}$ & $\cos{x}$ \\
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\midrule
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$\tan{x}$ & $1 + \tan^2{x} = \frac{1}{\cos^2{x}}$ \\
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\midrule
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$\cot{x}$ & $-1 - \cot^2{x} = \frac{-1}{\sin^2{x}}$ \\
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\midrule
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$\arccos{x}$ & $\frac{-1}{\sqrt{1 - x^2}}$ \\
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\midrule
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$\arcsin{x}$ & $\frac{1}{\sqrt{1 - x^2}}$ \\
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\midrule
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$\arctan{x}$ & $\frac{1}{1 + x^2}$ \\
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\bottomrule
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\end{tabularx}
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\paragraph{Équations différentielles}
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\columnbreak
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\begin{tabularx}{\linewidth}{lllc}
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\begin{tabularx}{\linewidth}{lY}
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\toprule
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\multicolumn{2}{l}{Type d'E.D.} & Solutions & \\
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\toprule
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\multicolumn{2}{l}{$ay' + by = 0$} & $\lambda e^{rx} \quad \text{ avec } r = \frac{-b}{a}$ & $a, b, \lambda\in\mathbb{R}$ \\
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\multirow{2}{*}{Linéarité} & $(u + v)' = u' + v'$ \\
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& $(au)' = au'$ \\
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\midrule
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\multicolumn{2}{l}{$ay' + by = f(x)$} & $y_0 + \lambda e^{rx} \quad \text{ avec } r = \frac{-b}{a}$ & \makecell{$y_0$ solution particulière de \\ $ay' + by = f(x)$ \\ $f$ une fonction et $a, b, \lambda\in\mathbb{R}$} \\
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Produit & $(uv)' = u'v + uv'$ \\
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\midrule
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\multirow{3}{*}{$ay'' + by' + cy = 0$} & $\Delta > 0$ & $\lambda e^{r_1 x} + \mu e^{r_2 x}$ & \multirowcell{3}[0pt][c]{$\lambda, \mu \in \mathbb{R}$ \\ $\alpha = \frac{-b}{2a} \quad \beta = \frac{\sqrt{|\Delta|}}{2a}$} \\
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\cline{2-3}
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& $\Delta = 0$ & $(\lambda x + \mu) e^{r_0 x}$ & \\
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\cline{2-3}
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& $\Delta < 0$ & $e^{\alpha x}(\lambda\cos{(\beta x)} + \mu\sin{(\beta x)})$ & \\
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Inverse & $\left(\frac{1}{v}\right)' = - \frac{v'}{v^2}$ \\
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\midrule
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Quotient & $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ \\
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\midrule
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Composée & $(f(u))' = u'f'(u)$ \\
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\bottomrule
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\end{tabularx}
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\paragraph{Solutions particulières des équations différentielles de 2\up{nd} ordre}
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\begin{tabularx}{\linewidth}{XX}
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\begin{tabularx}{\linewidth}{YY}
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\toprule
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\multicolumn{2}{c}{Second membre du type $e^{\alpha x}P(x)$} \\
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$\alpha$ non racine & $y_1 = e^{\alpha x} Q(x)$ \\
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$\alpha$ racine simple & $y_1 = x e^{\alpha x} Q(x)$ \\
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$\alpha$ racine double & $y_1 = x^2 e^{\alpha x} Q(x)$ \\
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Fonction & Primitive \\
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\toprule
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$u'u^n$ & $\frac{u^{n+1}}{n+1}$ \\
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\midrule
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\multicolumn{2}{c}{Second membre du type $e^{\alpha x}(P_1(x)\cos(\beta x) + P_2(x)\sin(\beta x))$} \\
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$\alpha + i\beta$ non racine & $y_1 = e^{\alpha x}(Q_1(x)\cos(\beta x) + Q_2(x)\sin(\beta x))$ \\
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$\alpha + i\beta$ racine & $y_1 = x e^{\alpha x}(Q_1(x)\cos(\beta x) + Q_2(x)\sin(\beta x))$ \\
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$\frac{u'}{u^2}$ & $-\frac{1}{u}$ \\
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\midrule
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$\frac{u'}{\sqrt{u}}$ & $2\sqrt{u}$ \\
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\midrule
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$u'\cos{u}$ & $\sin{u}$ \\
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\midrule
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$u'\sin{u}$ & $-\cos{u}$ \\
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\midrule
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$\frac{u'}{u}$ & $\ln{|u|}$ \\
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\midrule
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$u'e^u$ & $e^u$ \\
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\midrule
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$\frac{u'}{1 + u^2}$ & $\arctan{u}$ \\
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\bottomrule
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\end{tabularx}
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\paragraph{Intégrales généralisées}
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\end{multicols}
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Intégrales de référence~:
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\begin{tabular}{lll}
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\toprule
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Intégrale & converge si & diverge si \\
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\toprule
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$\int_a^{+\infty}\frac{1}{x^{\alpha}}\dif x$ & $\alpha > 1$ & $\alpha \leq 1$ \\
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$\int_a^{+\infty}e^{-\alpha x}\dif x$ & $\alpha > 0$ & $\alpha \leq 0$ \\
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$\int_a^{+\infty}x^n e^{-\alpha x}\dif x$ & $\alpha > 0$ & $\alpha \leq 0$ \\
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$\int_a^{+\infty}\frac{1}{x^{\alpha}(\ln x)^{\beta}}\dif x$ & $(\alpha > 1)$ ou $(\alpha = 1 \text{ et }\beta > 1)$ & $(\alpha < 1)$ ou $(\alpha = 1 \text{ et }\beta \leq 1)$ \\
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\bottomrule
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\end{tabular}
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\paragraph{Intégrales}\\
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$\int_a^b f(x)\dif x = [F(x)]_a^b = F(b) - F(a)$
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\begin{tabular}{|c|c|}
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\toprule
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IPP~: & changement de variables~: \\
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\midrule
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$\int_a^b uv'\dif x = [uv]_a^b - \int_a^b u'v\dif x$ & $\int_a^b f(x)\dif x = \int_{u(a)}^{u(b)} f(u)\frac{\dif u}{u'}$ \\
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\bottomrule
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\end{tabular}
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Majoration, minoration~:
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\begin{tabular}{lll}
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$0 \leq f(x) \leq g(x)$ & $\int_a^{+\infty}g(x)\dif x$ converge $\implies \int_a^{+\infty}f(x)\dif x$ converge aussi \\
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& $\int_a^{+\infty}f(x)\dif x$ diverge $\implies \int_a^{+\infty}g(x)\dif x$ diverge aussi \\
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\end{tabular}
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\paragraph{Équations différentielles}
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\paragraph{Séries de Fourier}
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$S_f(x) = a_0 + \sum_{n=1}^{+\infty}\left(a_n\cos{\frac{2\pi nx}{T}} + b_n\sin{\frac{2\pi nx}{T}}\right)$
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\begin{tabularx}{\linewidth}{lllc}
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\toprule
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\multicolumn{2}{l}{Type d'E.D.} & Solutions & \\
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\toprule
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\multicolumn{2}{l}{$ay' + by = 0$} & $\lambda e^{rx} \quad \text{ avec } r = \frac{-b}{a}$ & $a, b, \lambda\in\mathbb{R}$ \\
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\midrule
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\multicolumn{2}{l}{$ay' + by = f(x)$} & $y_0 + \lambda e^{rx} \quad \text{ avec } r = \frac{-b}{a}$ & \makecell{$y_0$ solution particulière de \\ $ay' + by = f(x)$ \\ $f$ une fonction et $a, b, \lambda\in\mathbb{R}$} \\
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\midrule
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\multirow{3}{*}{$ay'' + by' + cy = 0$} & $\Delta > 0$ & $\lambda e^{r_1 x} + \mu e^{r_2 x}$ & \multirowcell{3}[0pt][c]{$\lambda, \mu \in \mathbb{R}$ \\ $\alpha = \frac{-b}{2a} \quad \beta = \frac{\sqrt{|\Delta|}}{2a}$} \\
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\cline{2-3}
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& $\Delta = 0$ & $(\lambda x + \mu) e^{r_0 x}$ & \\
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\cline{2-3}
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& $\Delta < 0$ & $e^{\alpha x}(\lambda\cos{(\beta x)} + \mu\sin{(\beta x)})$ & \\
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\bottomrule
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\end{tabularx}
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avec
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\hfill
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$a_0 = \frac{1}{T} \int_{-L}^L f(x) \dif x$
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\hfill
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$a_n = \frac{2}{T} \int_{-L}^L f(x) \cos{\frac{2\pi nx}{T}} \dif x$
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\hfill
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$b_n = \frac{2}{T} \int_{-L}^L f(x) \sin{\frac{2\pi nx}{T}} \dif x$
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\paragraph{Solutions particulières des équations différentielles de 2\up{nd} ordre}
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$f$ paire $\implies b_n = 0$
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\qquad
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$f$ impaire $\implies a_0$ et $a_n = 0$
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\hfill
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$\cos(n\pi) = (-1)^n$
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\qquad
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$\sin(n\pi) = 0$
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\begin{tabularx}{\linewidth}{XX}
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\toprule
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\multicolumn{2}{c}{Second membre du type $e^{\alpha x}P(x)$} \\
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$\alpha$ non racine & $y_1 = e^{\alpha x} Q(x)$ \\
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$\alpha$ racine simple & $y_1 = x e^{\alpha x} Q(x)$ \\
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$\alpha$ racine double & $y_1 = x^2 e^{\alpha x} Q(x)$ \\
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\midrule
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\multicolumn{2}{c}{Second membre du type $e^{\alpha x}(P_1(x)\cos(\beta x) + P_2(x)\sin(\beta x))$} \\
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$\alpha + i\beta$ non racine & $y_1 = e^{\alpha x}(Q_1(x)\cos(\beta x) + Q_2(x)\sin(\beta x))$ \\
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$\alpha + i\beta$ racine & $y_1 = x e^{\alpha x}(Q_1(x)\cos(\beta x) + Q_2(x)\sin(\beta x))$ \\
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\bottomrule
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\end{tabularx}
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\paragraph{Intégrales généralisées}
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Intégrales de référence~:
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\begin{tabular}{lcc}
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\toprule
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Intégrale & converge si & diverge si \\
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\toprule
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$\int_a^{+\infty}\frac{1}{x^{\alpha}}\dif x$ & $\alpha > 1$ & $\alpha \leq 1$ \\
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$\int_a^{+\infty}e^{-\alpha x}\dif x$ & $\alpha > 0$ & $\alpha \leq 0$ \\
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$\int_a^{+\infty}x^n e^{-\alpha x}\dif x$ & $\alpha > 0$ & $\alpha \leq 0$ \\
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$\int_a^{+\infty}\frac{1}{x^{\alpha}(\ln x)^{\beta}}\dif x$ & $(\alpha > 1)$ ou $(\alpha = 1 \text{ et }\beta > 1)$ & $(\alpha < 1)$ ou $(\alpha = 1 \text{ et }\beta \leq 1)$ \\
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\bottomrule
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\end{tabular}
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Majoration, minoration~:
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\begin{tabular}{lll}
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$0 \leq f(x) \leq g(x)$ & $\int_a^{+\infty}g(x)\dif x$ converge $\implies \int_a^{+\infty}f(x)\dif x$ converge aussi \\
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& $\int_a^{+\infty}f(x)\dif x$ diverge $\implies \int_a^{+\infty}g(x)\dif x$ diverge aussi \\
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\end{tabular}
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\paragraph{Séries de Fourier}
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$S_f(x) = a_0 + \sum_{n=1}^{+\infty}\left(a_n\cos{\frac{2\pi nx}{T}} + b_n\sin{\frac{2\pi nx}{T}}\right)$
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avec
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\hfill
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$a_0 = \frac{1}{T} \int_{-L}^L f(x) \dif x$
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\hfill
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$a_n = \frac{2}{T} \int_{-L}^L f(x) \cos{\frac{2\pi nx}{T}} \dif x$
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\hfill
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$b_n = \frac{2}{T} \int_{-L}^L f(x) \sin{\frac{2\pi nx}{T}} \dif x$
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$f$ paire $\implies b_n = 0$ \\
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$f$ impaire $\implies a_0$ et $a_n = 0$
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\hfill
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$\cos(n\pi) = (-1)^n$
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\qquad
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$\sin(n\pi) = 0$
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\hfill{} \\
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Égalité de Parseval~:
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\hfill
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$\frac{1}{T}\int_{-L}^L f^2(x) \dif x = a_0^2 + \frac{1}{2}\sum_{n=1}^{+\infty}(a_n^2 + b_n^2)$
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\hfill{}
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\clearpage
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\section{Rappel sur les dérivées}
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Reference in a new issue