Quiz — Lycée Pilote Ariana - Exercice Intégrale et Primitive.pdf

Question 1 sur 10 20:00

[{"id":34010,"question":"Quelle est la primitive de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"A) x³ + x² - 5x + C","option_b":"B) x³ + x² - 5x","option_c":"C) 6x + 2 + C","option_d":"D) 3x³ + 2x² - 5x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 3x² est x³, celle de 2x est x², et celle de -5 est -5x. On ajoute la constante d'intégration C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A) x³ + x² - 5x + C\", \"b\": \"B) x³ + x² - 5x\", \"c\": \"C) 6x + 2","_debug_options_count":4},{"id":34011,"question":"L'intégrale définie ∫(de 0 à 1) (2x + 1) dx est égale à :","option_a":"A) 1","option_b":"B) 2","option_c":"C) 3","option_d":"D) 4","option_e":"","option_f":"","bonne_reponse":"c","explication":"La primitive de 2x + 1 est x² + x. Évaluée entre 0 et 1, cela donne (1 + 1) - (0 + 0) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A) 1\", \"b\": \"B) 2\", \"c\": \"C) 3\", \"d\": \"D) 4\"}}","_debug_options_count":4},{"id":34012,"question":"Vrai ou Faux ? La fonction F(x) = x² est une primitive de f(x) = 2x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de F(x) = x² est bien f(x) = 2x, donc F est une primitive de f.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":34013,"question":"Quelle technique utiliser pour calculer ∫ x·ln(x) dx ?","option_a":"A) Intégration par parties","option_b":"B) Changement de variable","option_c":"C) Décomposition en éléments simples","option_d":"D) Formule directe","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégration par parties est adaptée ici car on a un produit de x (facile à dériver) et ln(x) (facile à intégrer).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A) Intégration par parties\", \"b\": \"B) Changement de variable\", \"","_debug_options_count":4},{"id":34014,"question":"Vrai ou Faux ? ∫(de 0 à π) sin(x) dx = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La primitive de sin(x) est -cos(x). Évaluée entre 0 et π, on obtient -cos(π) - (-cos(0)) = 1 - (-1) = 2 ≠ 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":34015,"question":"Quelle est la valeur de ∫(de 1 à e) (1\/x) dx ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) e","option_d":"D) ln(e)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La primitive de 1\/x est ln(x). Évaluée entre 1 et e, cela donne ln(e) - ln(1) = 1 - 0 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A) 0\", \"b\": \"B) 1\", \"c\": \"C) e\", \"d\": \"D) ln(e)\"}}","_debug_options_count":4},{"id":34016,"question":"Vrai ou Faux ? Si F est une primitive de f, alors ∫(de a à b) f(x) dx = F(b) - F(a).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème fondamental de l'analyse qui énonce cette propriété essentielle des intégrales définies.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":34017,"question":"Quelle est la primitive de f(x) = e^(2x) ?","option_a":"A) e^(2x) + C","option_b":"B) (1\/2)e^(2x) + C","option_c":"C) 2e^(2x) + C","option_d":"D) e^(x) + C","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de (1\/2)e^(2x) est e^(2x), donc c'est une primitive de f(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A) e^(2x) + C\", \"b\": \"B) (1\/2)e^(2x) + C\", \"c\": \"C) 2e^(2x) + C\",","_debug_options_count":4},{"id":34018,"question":"Calculer ∫(de 0 à 1) x·e^(-x²) dx.","option_a":"A) 1 - 1\/e","option_b":"B) 1\/e","option_c":"C) e - 1","option_d":"D) 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Utilisez le changement de variable u = -x². La primitive est -(1\/2)e^(-x²), évaluée entre 0 et 1 donne 1\/2 - 1\/(2e).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A) 1 - 1\/e\", \"b\": \"B) 1\/e\", \"c\": \"C) e - 1\", \"d\": \"D) 0\"}}","_debug_options_count":4},{"id":34019,"question":"Vrai ou Faux ? Toute fonction continue admet une primitive sur son domaine de définition.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un résultat fondamental du calcul intégral : toute fonction continue sur un intervalle admet une primitive sur cet intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]