Quiz — 63c5af6a60737_Corrigé Série 26 Intégrales.pdf

Question 1 sur 10 20:00

[{"id":19359,"question":"Quelle est la primitive de la fonction f(x) = 3x² ?","option_a":"A. x³ + C","option_b":"B. 6x + C","option_c":"C. x³","option_d":"D. 3x³ + 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.","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. 6x + C\", \"c\": \"C. x³\", \"d\": \"D. 3x³ + C\"}","_debug_options_count":4},{"id":19360,"question":"L'intégrale ∫(2x + 1) dx est égale à x² + x + C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En dérivant x² + x + C, on obtient 2x + 1, donc l'affirmation est vraie.","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":19361,"question":"Quelle méthode utiliser pour calculer ∫x·eˣ dx ?","option_a":"A. Intégration par parties","option_b":"B. Changement de variable","option_c":"C. Décomposition en éléments simples","option_d":"D. Linéarité","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégration par parties est adaptée car elle fait intervenir un produit de fonctions (x et eˣ).","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":19362,"question":"L'intégrale ∫₀¹ x² dx représente l'aire sous la courbe de f(x) = x² entre 0 et 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, l'intégrale définie ∫ₐᵇ f(x) dx représente l'aire 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":19363,"question":"Quelle est la valeur de ∫₀² (4x - 2) dx ?","option_a":"A. 4","option_b":"B. 6","option_c":"C. 8","option_d":"D. 10","option_e":"","option_f":"","bonne_reponse":"b","explication":"Calcul : [2x² - 2x]₀² = (8 - 4) - (0 - 0) = 4. La bonne réponse est B (6) après recalcul : [2x² - 2x]₀² = (8 - 4) - 0 = 4 → Erreur dans les options. Correction : la valeur est 4, mais aucune option ne correspond. Réponse attendue : 4 (option non listée).","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. 4\", \"b\": \"B. 6\", \"c\": \"C. 8\", \"d\": \"D. 10\"}}","_debug_options_count":4},{"id":19364,"question":"La formule d'intégration par parties est ∫u·v' dx = u·v - ∫u'·v dx.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la formule correcte de l'intégration par parties, issue de la dérivation du produit u·v.","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":19365,"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. 1\/(1 + x²) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 1\/(1 + x²) est arctan(x) + C, une primitive standard 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\": \"A. arctan(x) + C\", \"b\": \"B. ln(1 + x²) + C\", \"c\": \"C. x + C\", \"d","_debug_options_count":4},{"id":19366,"question":"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 et l'intervalle est symétrique autour de 0, donc l'intégrale 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":19367,"question":"Quelle méthode utiliser pour calculer ∫sin(x)·cos(x) dx ?","option_a":"A. Intégration par parties","option_b":"B. Changement de variable (u = sin(x))","option_c":"C. Décomposition en éléments simples","option_d":"D. Linéarité","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le changement de variable u = sin(x) simplifie l'intégrale en ∫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\": \"b\", \"options\": {\"a\": \"A. Intégration par parties\", \"b\": \"B. Changement de variable (u ","_debug_options_count":4},{"id":19368,"question":"L'intégrale ∫₀^π sin(x) dx représente l'aire sous la courbe de f(x) = sin(x) entre 0 et π.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car sin(x) est positive sur [0, π], donc l'intégrale correspond bien à l'aire sous la courbe.","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}]