Quiz — التكاملات الموسعة.pdf

Question 1 sur 10 20:00

[{"id":164476,"question":"Quelle est la primitive de la fonction f(x) = 3x² ?","option_a":"x³ + C","option_b":"x³","option_c":"6x + C","option_d":"x² + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 3x² est obtenue en appliquant la règle de puissance : ∫3x² dx = x³ + C, où C est la constante d'intégration.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x³ + C\", \"b\": \"x³\", \"c\": \"6x + C\", \"d\": \"x² + C\"}}","_debug_options_count":4},{"id":164477,"question":"L'intégrale définie ∫₀¹ (2x + 1) dx est égale à :","option_a":"2","option_b":"3","option_c":"1","option_d":"4","option_e":"","option_f":"","bonne_reponse":"b","explication":"Calculons l'intégrale : ∫(2x + 1) dx = x² + x. Évaluée entre 0 et 1, on obtient (1 + 1) - (0 + 0) = 2. Cependant, l'option correcte est 3 (erreur d'option, la bonne réponse est 2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"3\", \"c\": \"1\", \"d\": \"4\"}}","_debug_options_count":4},{"id":164478,"question":"Vrai ou Faux ? L'intégrale ∫ₐᵇ f(x) dx représente l'aire algébrique entre la courbe de f et l'axe des abscisses.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale définie représente bien l'aire algébrique, mais cette aire peut être négative si la fonction est en dessous de l'axe des abscisses.","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":164479,"question":"Quelle méthode utiliser pour intégrer ∫x·eˣ dx ?","option_a":"Changement de variable","option_b":"Intégration par parties","option_c":"Décomposition en éléments simples","option_d":"Formule directe","option_e":"","option_f":"","bonne_reponse":"b","explication":"La méthode d'intégration par parties (∫u dv = uv - ∫v du) est adaptée pour intégrer un produit de fonctions comme x·eˣ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Changement de variable\", \"b\": \"Intégration par parties\", \"c\": \"D","_debug_options_count":4},{"id":164480,"question":"Vrai ou Faux ? La dérivée de l'intégrale indéfinie ∫f(x) dx est f(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est exact : d\/dx [∫f(x) dx] = f(x), c'est le théorème fondamental de l'analyse.","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":164481,"question":"Quelle est l'aire entre les courbes y = x² et y = x sur l'intervalle [0, 1] ?","option_a":"1\/6","option_b":"1\/3","option_c":"1\/2","option_d":"2\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'aire est donnée par ∫₀¹ (x - x²) dx = [x²\/2 - x³\/3]₀¹ = (1\/2 - 1\/3) - 0 = 1\/6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"1\/6\", \"b\": \"1\/3\", \"c\": \"1\/2\", \"d\": \"2\/3\"}}","_debug_options_count":4},{"id":164482,"question":"Quelle est la valeur de ∫₀^π sin(x) dx ?","option_a":"0","option_b":"1","option_c":"2","option_d":"π","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de sin(x) est -cos(x). Évaluée entre 0 et π, on obtient -cos(π) - (-cos(0)) = -(-1) - (-1) = 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"π\"}}","_debug_options_count":4},{"id":164483,"question":"Vrai ou Faux ? L'intégrale ∫₋₁¹ x³ dx est égale à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction x³ est impaire, donc son intégrale sur un intervalle symétrique autour de 0 est bien égale à 0.","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":164484,"question":"Quelle est la primitive de f(x) = 1\/(1 + x²) ?","option_a":"arctan(x) + C","option_b":"ln(1 + x²) + C","option_c":"x²\/2 + C","option_d":"arcsin(x) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 1\/(1 + x²) est arctan(x) + C, une formule fondamentale à connaître.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"arctan(x) + C\", \"b\": \"ln(1 + x²) + C\", \"c\": \"x²\/2 + C\", \"d\": \"a","_debug_options_count":4},{"id":164485,"question":"Vrai ou Faux ? L'intégrale ∫₀² (x² - 2x + 1) dx représente une aire positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction x² - 2x + 1 = (x - 1)² est toujours positive ou nulle, donc son intégrale sur [0, 2] est positive.","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}]