31 lines
984 B
Markdown
31 lines
984 B
Markdown
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alias: [ "intégrale de 1/(x²+a²)" ]
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---
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up:: [[intégration]]
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title:: "$\displaystyle \int \frac{1}{x^{2}+a^{2}} \, dx = \frac{1}{a}\arctan\left( \frac{x}{a} \right)$"
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#s/maths/analyse
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---
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# Généralisation
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## composée avec une fonction quelconque
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$\big( \arctan(u) \big)' = u' \arctan'(u) = \frac{u'}{1+u^{2}}$
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donc :
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$\boxed{\displaystyle\int \frac{u'}{1+u^{2}} \, dx = \arctan(u)}$
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## facteur devant le $x$
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Plus généralement, on obtient
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$\displaystyle \int \frac{1}{(kx)^{2} + a^{2}} \, dx = \frac{1}{ka} \arctan\left( \frac{xk}{a} \right)$
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> [!definition] démonstration
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> $\displaystyle\int \frac{1}{a^{2}+(kx)^{2}} \, dx = \frac{1}{k^{2}}\int \frac{1}{ \frac{a^{2}}{k^{2}} +x^{2}} \, dx = \frac{1}{k^{2}}\times \frac{k}{a} \arctan\left( \frac{xk}{a} \right) = \boxed{\frac{1}{ka} \arctan\left( \frac{xk}{a} \right)}$
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> [!example] Exemple
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> $\displaystyle \int \frac{1}{1+2x^{2}} \, dx = \frac{1}{\sqrt{ 2 }} \arctan\left( \frac{x}{\sqrt{ 2 }} \right)$
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