Quiz — Intégration des fonctions numériques.pdf

Question 1 sur 10 20:00

[{"id":47468,"question":"Quelle est la primitive de la fonction f(x) = 3x² ?","option_a":"x³ + C","option_b":"x³","option_c":"3x³ + C","option_d":"x³\/3 + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de x^n est x^(n+1)\/(n+1) + C. Ici, n=2, donc la primitive est 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\": \"x³ + C\", \"b\": \"x³\", \"c\": \"3x³ + C\", \"d\": \"x³\/3 + C\"}}","_debug_options_count":4},{"id":47469,"question":"L'intégrale ∫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 : [x² + x]₀¹ = (1 + 1) - (0 + 0) = 2. L'intégrale vaut donc 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":47470,"question":"Vrai ou Faux : L'intégrale d'une fonction paire sur [-a, a] est toujours nulle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. L'intégrale d'une fonction paire sur [-a, a] est égale à 2 fois son intégrale sur [0, a]. Elle est nulle uniquement si la fonction est nulle sur [0, a].","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":47471,"question":"Quelle méthode utiliser pour calculer ∫x·e^x dx ?","option_a":"Substitution simple","option_b":"Intégration par parties","option_c":"Décomposition en éléments simples","option_d":"Changement de variable","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégration par parties est adaptée ici car le produit de x (polynôme) et e^x (exponentielle) ne peut pas être intégré directement.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Substitution simple\", \"b\": \"Intégration par parties\", \"c\": \"Déc","_debug_options_count":4},{"id":47472,"question":"L'intégrale ∫₀^π sin(x) dx représente :","option_a":"L'aire sous la courbe de sin(x) entre 0 et π","option_b":"La longueur de l'arc de sin(x) entre 0 et π","option_c":"Le volume engendré par sin(x)","option_d":"La dérivée de sin(x)","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\": \"L'aire sous la courbe de sin(x) entre 0 et π\", \"b\": \"La longueur","_debug_options_count":4},{"id":47473,"question":"Vrai ou Faux : La primitive de 1\/x est ln|x| + C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. La dérivée de ln|x| est 1\/x, donc ln|x| + C est bien une primitive de 1\/x.","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":47474,"question":"Quelle est la valeur de ∫₁² (1\/x) dx ?","option_a":"ln(2)","option_b":"ln(2) - ln(1)","option_c":"1\/2","option_d":"2","option_e":"","option_f":"","bonne_reponse":"b","explication":"∫₁² (1\/x) dx = [ln|x|]₁² = ln(2) - ln(1) = ln(2) - 0 = ln(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\": \"ln(2)\", \"b\": \"ln(2) - ln(1)\", \"c\": \"1\/2\", \"d\": \"2\"}}","_debug_options_count":4},{"id":47475,"question":"La méthode de substitution est utile pour intégrer des fonctions composées comme :","option_a":"x² + 1","option_b":"e^(2x)","option_c":"x·ln(x)","option_d":"sin(3x)","option_e":"","option_f":"","bonne_reponse":"d","explication":"La substitution est idéale pour les fonctions composées comme sin(3x), où on pose u = 3x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"x² + 1\", \"b\": \"e^(2x)\", \"c\": \"x·ln(x)\", \"d\": \"sin(3x)\"}}","_debug_options_count":4},{"id":47476,"question":"Vrai ou Faux : Toute fonction continue admet une primitive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. D'après le théorème fondamental de l'analyse, 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},{"id":47477,"question":"Quelle est la primitive de f(x) = e^(2x) ?","option_a":"e^(2x)\/2 + C","option_b":"2e^(2x) + C","option_c":"e^(2x) + C","option_d":"e^(2x+1)\/2 + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de e^(kx) est e^(kx)\/k + C. Ici, k=2, donc la primitive est e^(2x)\/2 + 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\": \"e^(2x)\/2 + C\", \"b\": \"2e^(2x) + C\", \"c\": \"e^(2x) + C\", \"d\": \"e^(2x","_debug_options_count":4}]