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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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\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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@ -27,14 +26,13 @@
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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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\paragraph{Dérivées et Primitives}
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\paragraph{Dérivées et Primitives}
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\begin{multicols}{2}
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@ -42,31 +40,31 @@
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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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$a$ & 0 \\
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\midrule
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\textcolor{red}{$ax$} & \textcolor{red}{$a$} \\
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$ax$ & $a$ \\
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\midrule
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$\frac{1}{2} x^2$ & \textcolor{red}{$x$} \\
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$\frac{1}{2} x^2$ & $x$ \\
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\midrule
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\textcolor{red}{$x^n$} & \textcolor{red}{$nx^{n-1}$} \\
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$x^n$ & $nx^{n-1}$ \\
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\midrule
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\textcolor{red}{$\sqrt{x}$} & $\frac{1}{2\sqrt{x}}$ \\
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$\sqrt{x}$ & $\frac{1}{2\sqrt{x}}$ \\
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\midrule
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$\frac{2}{3} x\sqrt{x}$ & \textcolor{red}{$\sqrt{x}$} \\
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$\frac{2}{3} x\sqrt{x}$ & $\sqrt{x}$ \\
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\midrule
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\textcolor{red}{$e^{ax}$} & \textcolor{red}{$ae^{ax}$} \\
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$e^{ax}$ & $ae^{ax}$ \\
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\midrule
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\textcolor{red}{$a^x$} & $a^x \ln{a}$ \\
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$a^x$ & $a^x \ln{a}$ \\
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\midrule
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\textcolor{red}{$\ln{|x|}$} & \textcolor{red}{$\frac{1}{x}$} \\
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$\ln{|x|}$ & $\frac{1}{x}$ \\
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\midrule
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\textcolor{red}{$-\frac{1}{x}$} & \textcolor{red}{$\frac{1}{x^2}$} \\
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$-\frac{1}{x}$ & $\frac{1}{x^2}$ \\
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\midrule
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\textcolor{red}{$\cos{x}$} & \textcolor{red}{$-\sin{x}$} \\
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$\cos{x}$ & $-\sin{x}$ \\
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\midrule
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\textcolor{red}{$\sin{x}$} & \textcolor{red}{$\cos{x}$} \\
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$\sin{x}$ & $\cos{x}$ \\
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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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$\tan{x}$ & $1 + \tan^2{x} = \frac{1}{\cos^2{x}}$ \\
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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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@ -74,7 +72,7 @@
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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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$\arctan{x}$ & $\frac{1}{1 + x^2}$ \\
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\bottomrule
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\end{tabularx}
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@ -119,18 +117,17 @@
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\end{multicols}
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\paragraph{Intégrales}
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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{tabularx}{\linewidth}{YY}
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\begin{tabular}{|c|c|}
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\toprule
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Intégration par parties~: & Intégration par changement de variables~: \\
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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{tabularx}
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\end{tabular}
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\paragraph{Équations différentielles}
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\paragraph{Équations différentielles}
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\begin{tabularx}{\linewidth}{lllc}
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\toprule
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@ -148,7 +145,7 @@
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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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\paragraph{Solutions particulières des équations différentielles de 2\up{nd} ordre}
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\begin{tabularx}{\linewidth}{XX}
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\toprule
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@ -163,10 +160,10 @@
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\bottomrule
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\end{tabularx}
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\paragraph{Intégrales généralisées}
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\paragraph{Intégrales généralisées}
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Intégrales de référence~:
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\begin{tabular}{lll}
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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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@ -183,7 +180,7 @@
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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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\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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@ -194,13 +191,18 @@
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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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\qquad
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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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