diff --git a/analyse/main.tex b/analyse/main.tex index 8c16020..d4e0f9e 100644 --- a/analyse/main.tex +++ b/analyse/main.tex @@ -274,12 +274,15 @@ \paragraph{Exemple} \begin{align*} - &\int_0^{\ln{\sqrt{3}}} \frac{1}{e^x + e^{-x}}\,\mathrm{d}x\text{, en posant }u=e^x \\\\ - &= \int_{e^0}^{e^{\ln{\sqrt{3}}}} \frac{1}{u + \frac{1}{u}}\,\frac{\mathrm{d}u}{u} - = \int_1^{\sqrt{3}} \frac{1}{u^2 + 1}\,\mathrm{d}u \\ - &= [\arctan{u}]_1^{\sqrt{3}} - = \arctan{\sqrt{3}} - \arctan{1} \\ - &= \frac{\pi}{3} - \frac{\pi}{4} = \frac{\pi}{12} + \int_0^{\ln{\sqrt{3}}} \frac{1}{e^x + e^{-x}}\,\mathrm{d}x\text{,\quad on pose }u=e^x + \end{align*} + + \begin{align*} + &= \int_{e^0}^{e^{\ln{\sqrt{3}}}} \frac{1}{u + \frac{1}{u}}\,\frac{\mathrm{d}u}{u} \\\\ + &= \int_1^{\sqrt{3}} \frac{1}{u^2 + 1}\,\mathrm{d}u + = [\arctan{u}]_1^{\sqrt{3}} + = \arctan{\sqrt{3}} - \arctan{1} + = \frac{\pi}{3} - \frac{\pi}{4} = \frac{\pi}{12} \end{align*} \section{Equations différentielles}