Montrer que pour tout réel x :
\(\displaystyle{\dfrac{2}{e^x+2}=1-\dfrac{e^x}{e^x+2}}\)
\(\displaystyle{\dfrac{e^x}{e^x+2}-1=\dfrac{e^x}{e^x+2}-\dfrac{e^x+2}{e^x+2}=\dfrac{e^x-e^x+2}{e^x+2}=\dfrac{2}{e^x+2}}\)
Pour tout réel \(\displaystyle{x}\), \(\displaystyle{1-\dfrac{e^x}{e^x+2}=\dfrac{e^x+2}{e^x+2}-\dfrac{e^x}{e^x+2}=\dfrac{\left(e^x+2\right)-e^x}{e^x+2}=\dfrac{2}{e^x+2}}\)
En déduire la valeur de :
\(\displaystyle{A=\int_{0}^{ln2} \dfrac{2}{e^x+2} \ \mathrm dx}\)
\(\displaystyle{A=e^{\frac{3}{2}}}\)
\(\displaystyle{A=\dfrac{3}{2}}\)
\(\displaystyle{A=\ln\left(\dfrac{2}{3}\right)}\)
\(\displaystyle{A=\ln\left(\dfrac{3}{2}\right)}\)
Sommaire du chapitre
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