Quiz — série primitives et calcul intégral.pdf

Question 1 sur 10 20:00

[{"id":123678,"question":"Quelle est la primitive de la fonction f(x) = 3x² ?","option_a":"A. x³ + C","option_b":"B. x³","option_c":"C. 6x + C","option_d":"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 = 3*(x³\/3) + C = x³ + 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 + C\", \"b\": \"B. x³\", \"c\": \"C. 6x + C\", \"d\": \"D. x² + C\"}}","_debug_options_count":4},{"id":123679,"question":"L'intégrale ∫₀¹ eˣ dx est égale à :","option_a":"A. e - 1","option_b":"B. e","option_c":"C. 1","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de eˣ est eˣ, donc ∫₀¹ eˣ dx = e¹ - e⁰ = e - 1.","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. e - 1\", \"b\": \"B. e\", \"c\": \"C. 1\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":123680,"question":"Vrai ou Faux : Toute fonction continue sur un intervalle admet une primitive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème fondamental de l'analyse : si f est continue sur [a,b], alors f 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},{"id":123681,"question":"Quelle méthode utiliser pour calculer ∫x*eˣ dx ?","option_a":"A. Intégration directe","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 dv = uv - ∫v du) est adaptée ici avec u = x et dv = eˣ dx.","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. Intégration directe\", \"b\": \"B. Intégration par parties\", \"c\"","_debug_options_count":4},{"id":123682,"question":"L'intégrale ∫₀^π sin(x) dx représente :","option_a":"A. L'aire sous la courbe de sin(x) de 0 à π","option_b":"B. La longueur de l'arc de sin(x)","option_c":"C. Le volume sous la courbe de sin(x)","option_d":"D. Aucune des réponses","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une intégrale définie ∫ₐᵇ f(x) dx représente l'aire algébrique sous la courbe de f entre a et b.","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. L'aire sous la courbe de sin(x) de 0 à π\", \"b\": \"B. La longu","_debug_options_count":4},{"id":123683,"question":"Vrai ou Faux : ∫(f(x) + g(x)) dx = ∫f(x) dx + ∫g(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 de l'intégrale : l'intégrale d'une somme est la somme des intégrales.","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":123684,"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 + C","option_d":"D. arccos(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,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. arctan(x) + C\", \"b\": \"B. ln(1+x²) + C\", \"c\": \"C. x + C\", \"d\":","_debug_options_count":4},{"id":123685,"question":"L'intégrale ∫₀² (2x + 1) dx est égale à :","option_a":"A. 6","option_b":"B. 4","option_c":"C. 8","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"∫(2x + 1) dx = x² + x, donc ∫₀² (2x + 1) dx = (4 + 2) - (0 + 0) = 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\": \"A. 6\", \"b\": \"B. 4\", \"c\": \"C. 8\", \"d\": \"D. 2\"}}","_debug_options_count":4},{"id":123686,"question":"Vrai ou Faux : Si F est une primitive de f, alors ∫ₐᵇ 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 : l'intégrale définie se calcule à partir des primitives.","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":123687,"question":"Quelle est la valeur de ∫₀^π\/2 cos(x) dx ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. π\/2","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La primitive de cos(x) est sin(x), donc ∫₀^π\/2 cos(x) dx = sin(π\/2) - sin(0) = 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. π\/2\", \"d\": \"D. -1\"}}","_debug_options_count":4}]