Réponse :
Explications étape par étape
Bonsoir
f(x) = a / (x - 2) + b / (x - 3)
f(x) = (x - 5) / (x^2 - 5x + 6)
f(x) = [a(x - 3) + b(x - 2)]/[(x - 2)(x - 3)]
f(x) = (ax - 3a + bx - 2b)/[(x - 2)(x - 3)]
(x - 2)(x - 3) = x^2 - 3x - 2x + 6
(x - 2)(x - 3) = x^2 - 5x + 6
Donc on a :
ax - 3a + bx - 2b = x - 5
x(a + b) - 3a - 2b = x - 5
a + b = 1
-3a - 2b = -5
a = 1 - b
-3(1 - b) - 2b = -5
-3 + 3b - 2b = -5
b = -5 + 3
b = -2
a = 1 - (-2) = 1 + 2 = 3
f(x) = 3/(x - 2) - 2/(x - 3)
2) primitive de f :
f(x) = 3u’/u - 2v’/v
F(x) = 3 Ln |u| - 2 Ln |v|
F(x) = 3 Ln |x - 2| - 2 Ln |x - 3|
