Mathc initiation/Fichiers c : c78di2
Calculons la primitive :
/
Calculer la primitive de | cosh(x)**n dx
/
Utilisons l'intégration par partie
u = ... dv = ...
du = ... v = ...
/ /
| u dv = u v - | v du
/ /
/
| cosh(x)**n dx =
/
u = cosh(x)**(n-1) dv = cosh(x) dx
du = (n-1) cosh(x)**(n-2) sinh(x) dx v = sinh(x)
(u dv) (u*v) (v*du)
/ /
| cosh(x)**n dx = cosh(x)**(n-1) * sinh(x) - | sinh(x) * (n-1) cos(x)**(n-2) sin(x) dx
/ /
/ /
| cosh(x)**n dx = cosh(x)**(n-1) sinh(x) - (n-1) | sinh(x)**2 cos(x)**(n-2) dx
/ /
sinh(x)**2 = (cosh(x)**2 - 1)
/ /
| cosh(x)**n dx = cosh(x)**(n-1) sinh(x) - (n-1) | (cosh(x)**2 - 1) cosh(x)**(n-2) dx
/ /
/ /
| cosh(x)**n dx = cosh(x)**(n-1) sinh(x) - (n-1) | (cosh(x)**n - cosh(x)**(n-2)) dx
/ /
/ / /
| cosh(x)**n dx = cosh(x)**(n-1) sinh(x) - (n-1) | cosh(x)**n dx + (n-1) | cosh(x)**(n-2) dx
/ / /
/ /
n | cosh(x)**n dx = cosh(x)**(n-1) sinh(x) + (n-1) | cosh(x)**(n-2) dx
/ /
/ 1 (n-1) /
| cosh(x)**n dx = --- cosh(x)**(n-1) sinh(x) + ----- | cosh(x)**(n-2) dx
/ n n /