Testez vos connaissances sur les intégrales — Terminale Mathématiques

Question 1 sur 10 10:00

[{"id":42886,"question":"Quelle est la primitive de la fonction f(x) = 3x² + 2x - 1 ?","option_a":"A) x³ + x² - x + C","option_b":"B) x³ + x² - x","option_c":"C) 6x + 2 + C","option_d":"D) 3x³ + 2x² - x + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive s'obtient en intégrant terme à terme : ∫3x² dx = x³, ∫2x dx = x², ∫-1 dx = -x. La constante C est ajoutée pour les primitives.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42887,"question":"L'intégrale ∫₀¹ (2x + 1) dx est égale à :","option_a":"A) 2","option_b":"B) 1","option_c":"C) 3","option_d":"D) 0","option_e":"","option_f":"","bonne_reponse":"A","explication":"Calculons la primitive F(x) = x² + x. Puis F(1) - F(0) = (1+1) - (0+0) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42888,"question":"La relation de Chasles stipule que ∫ₐᵇ f(x) dx + ∫ᵇᶜ f(x) dx = ∫ₐᶜ f(x) dx.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est une propriété fondamentale des intégrales : l'additivité par rapport aux bornes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42889,"question":"Quelle méthode utiliser pour calculer ∫x eˣ dx ?","option_a":"A) Primitive immédiate","option_b":"B) Intégration par parties","option_c":"C) Changement de variable","option_d":"D) Décomposition en éléments simples","option_e":"","option_f":"","bonne_reponse":"B","explication":"La méthode d'intégration par parties (u = x, dv = eˣ dx) est adaptée aux produits de fonctions polynomiales et exponentielles.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42890,"question":"L'aire entre la courbe y = x² et l'axe des abscisses sur [0,2] vaut :","option_a":"A) 8\/3","option_b":"B) 4","option_c":"C) 2","option_d":"D) 16\/3","option_e":"","option_f":"","bonne_reponse":"A","explication":"Calculons ∫₀² x² dx = [x³\/3]₀² = 8\/3. L'aire est positive car la fonction est au-dessus de l'axe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42891,"question":"Si f est une fonction paire, alors ∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par symétrie, l'intégrale sur [-a,a] d'une fonction paire est le double de l'intégrale sur [0,a].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42892,"question":"Quelle est la valeur de ∫₀¹ (3x² + 2) dx ?","option_a":"A) 3","option_b":"B) 4","option_c":"C) 5","option_d":"D) 6","option_e":"","option_f":"","bonne_reponse":"B","explication":"Primitive F(x) = x³ + 2x. F(1) - F(0) = (1+2) - 0 = 3. Attention à l'option B : 4 est incorrect.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42893,"question":"L'intégrale ∫₁ᵉ ln(x)\/x dx vaut :","option_a":"A) (ln(e))²\/2","option_b":"B) 1\/2","option_c":"C) (ln(1))²\/2","option_d":"D) e²\/2","option_e":"","option_f":"","bonne_reponse":"A","explication":"Posons u = ln(x), du = dx\/x. L'intégrale devient ∫₀¹ u du = [u²\/2]₀¹ = 1\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42894,"question":"La méthode de décomposition en éléments simples s'applique aux intégrales de fractions rationnelles.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Cette méthode permet de décomposer une fraction rationnelle en termes plus simples avant intégration.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42895,"question":"Quelle est la primitive de f(x) = 1\/(1+x²) ?","option_a":"A) arctan(x) + C","option_b":"B) ln(1+x²) + C","option_c":"C) x\/(1+x²) + C","option_d":"D) arcsin(x) + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de arctan(x) est 1\/(1+x²), donc sa primitive est arctan(x) + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]