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règles de Bioche :
Soit \(\omega(x)=f(x)dx\)
Alors \(\omega(-x)=f(-x)d(-x)\)- \(\omega(-x)=\omega(x)\) \(\to\) changement de variable \(u=\cos x\)
- \(\omega(\pi-x)=\omega (x)\) \(\to\) changement de variable \(u=\sin x\)
- \(\omega(\pi+x)=\omega(x)\) \(\to\) changement de variable \(u=\tan x\)
Exemple :
Soit $$F=\int{\cos x\over2-\cos^2x}dx$$
On a alors$$\omega(x)={\cos x\over2-\cos^2x}dx$$
Ici $$\begin{align}\omega(\pi-x)&={\cos (\pi-x)\over2-\cos^2(\pi-x)}dx\\ &=\omega(x)\end{align}$$
Soit \(u=\sin x\), \(du=\cos x\,dx\)
$$\begin{align}F&=\int{\cos x\,dx\over2-(1-\sin^2x)}\\ &=\int{du\over 1+u^2}\\ &=\arctan u+k,k\in\Bbb R\\ &=\arctan(\sin x)+k\end{align}$$
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Rétroliens :