Bonjour,
f(x) = 1 - (x +1)e⁻ˣ
1) f'(x) = -1*e⁻ˣ - (x + 1)*(-e⁻ˣ) = xe⁻ˣ
2) Sur [0;6], x ≥ 0 ⇒ f'(x) ≥ 0
x 0 6
f'(x) 0 +
f(x) 0 croissante 1-5/e⁶
3)a) F(x) = x + (x + 2)e⁻ˣ
⇒ F'(x) = 1 + e⁻ˣ - (x + 2)eˣ = 1 - (x + 1)e⁻ˣ = f(x)
⇒ F primitive de f
b) I = F(6) - F(0)
= (6 + 8e⁻⁶) - (2)
= 4 + 8e⁻⁶
≈ 4,0198
