Mathc initiation/Fichiers c : c77di2
Apparence
Calculons la primitive :
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Calculer la primitive de | cos(x)**n dx
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Utilisons l'intégration par partie
u = ... dv = ...
du = ... v = ...
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| u dv = u v - | v du
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| cos(x)**n dx =
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u = cos(x)**(n-1) dv = cos(x) dx
du = (n-1) cos(x)**(n-2) (-sin(x) dx v = sin(x)
(u dv) (u*v) (v*du)
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| cos(x)**n dx = cos(x)**(n-1) * sin(x) - | sin(x) * (n-1) cos(x)**(n-2) (-sin(x)) dx
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| cos(x)**n dx = cos(x)**(n-1) * sin(x) + (n-1) | sin(x)**2 cos(x)**(n-2) dx
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sin(x)**2 = (1-cos(x)**2)
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| cos(x)**n dx = cos(x)**(n-1) * sin(x) + (n-1) | (1-cos(x)**2) cos(x)**(n-2) dx
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| cos(x)**n dx = cos(x)**(n-1) * sin(x) + (n-1) | (cos(x)**(n-2) - cos(x)**n) dx
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| cos(x)**n dx = cos(x)**(n-1) * sin(x) + (n-1) | cos(x)**(n-2) dx - (n-1) | cos(x)**n dx
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| cos(x)**n dx = cos(x)**(n-1) * sin(x) + (n-1) | cos(x)**(n-2) dx - n | cos(x)**n dx + | cos(x)**n dx
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n | cos(x)**n dx = cos(x)**(n-1) * sin(x) + (n-1) | cos(x)**(n-2) dx
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/ 1 (n-1) /
| cos(x)**n dx = --- cos(x)**(n-1) * sin(x) + ----- | cos(x)**(n-2) dx
/ n n /