Quiz — الدوال الأصلية والتكامل تمارين محلولة.pdf

Question 1 sur 10 20:00

[{"id":13664,"question":"Quelle est la primitive de la fonction f(x) = 3x² + 2x - 1 ?","option_a":"F(x) = x³ + x² - x + C","option_b":"F(x) = 6x + 2 + C","option_c":"F(x) = x³ + x - x² + C","option_d":"F(x) = 3x³ + 2x² - x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive d'une fonction polynomiale s'obtient en augmentant chaque exposant de 1 et en divisant par le nouvel exposant. Ici, 3x² devient x³, 2x devient x², et -1 devient -x. La constante C est ajoutée car la primitive n'est pas unique.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"F(x) = x³ + x² - x + C\", \"b\": \"F(x) = 6x + 2 + C\", \"c\": \"F(x) =","_debug_options_count":4},{"id":13665,"question":"L'intégrale définie ∫₀¹ (2x + 1) dx est égale à :","option_a":"1","option_b":"2","option_c":"1.5","option_d":"3","option_e":"","option_f":"","bonne_reponse":"c","explication":"Calculons l'intégrale : ∫(2x + 1) dx = x² + x + C. Évaluons entre 0 et 1 : (1² + 1) - (0² + 0) = 2. L'intégrale définie donne donc 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\": \"1\", \"b\": \"2\", \"c\": \"1.5\", \"d\": \"3\"}}","_debug_options_count":4},{"id":13666,"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":"C'est un théorème fondamental de l'analyse : si une fonction est continue sur un intervalle, alors elle 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":13667,"question":"Quelle méthode utiliser pour calculer ∫ sin(x) cos(x) dx ?","option_a":"Substitution u = sin(x)","option_b":"Intégration par parties","option_c":"Décomposition en éléments simples","option_d":"Changement de variable x = tan(t)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La substitution u = sin(x) fonctionne car du = cos(x) dx. L'intégrale devient ∫ u du = u²\/2 + C = sin²(x)\/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\": \"Substitution u = sin(x)\", \"b\": \"Intégration par parties\", \"c\": \"","_debug_options_count":4},{"id":13668,"question":"L'intégrale ∫₀²π sin(x) dx est égale à :","option_a":"0","option_b":"1","option_c":"2","option_d":"π","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction sin(x) est périodique de période 2π et symétrique par rapport à l'origine sur [0, 2π]. L'aire positive compense exactement l'aire négative, donc l'intégrale est nulle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"π\"}}","_debug_options_count":4},{"id":13669,"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":"C'est une primitive fondamentale : d\/dx [ln|x|] = 1\/x pour x ≠ 0. Attention, la fonction 1\/x n'est pas définie en x=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":13670,"question":"Quelle est la valeur de ∫₀¹ eˣ dx ?","option_a":"e - 1","option_b":"1","option_c":"e","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de eˣ est eˣ + C. Évaluons entre 0 et 1 : 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\": \"e - 1\", \"b\": \"1\", \"c\": \"e\", \"d\": \"0\"}}","_debug_options_count":4},{"id":13671,"question":"Vrai ou Faux : ∫ₐᵇ f(x) dx = -∫ᵇₐ 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'intégrale change de signe lorsque les bornes sont inversées.","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":13672,"question":"Quelle est la primitive de f(x) = 1\/(1 + x²) ?","option_a":"arctan(x) + C","option_b":"ln(1 + x²) + C","option_c":"arcsin(x) + C","option_d":"1\/(1 + x²) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de arctan(x) est 1\/(1 + x²), donc arctan(x) est une primitive de cette fonction.","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\": \"arcsin(x) + C\", \"d\"","_debug_options_count":4},{"id":13673,"question":"L'intégrale ∫₀¹ x eˣ dx est égale à :","option_a":"e - 1","option_b":"1","option_c":"e","option_d":"e - 2","option_e":"","option_f":"","bonne_reponse":"d","explication":"Utilisons l'intégration par parties : u = x, dv = eˣ dx → du = dx, v = eˣ. L'intégrale devient [x eˣ]₀¹ - ∫₀¹ eˣ dx = (e - 0) - (e - 1) = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"e - 1\", \"b\": \"1\", \"c\": \"e\", \"d\": \"e - 2\"}}","_debug_options_count":4}]