Quiz — Lycée Pilote Bizerte - Intégrales 2.pdf

Question 1 sur 10 20:00

[{"id":8315,"question":"Quelle est la primitive de la fonction f(x) = 3x² ?","option_a":"A. x³","option_b":"B. x³ + C","option_c":"C. 6x","option_d":"D. x² + C","option_e":"","option_f":"","bonne_reponse":"b","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\": \"b\", \"options\": {\"a\": \"A. x³\", \"b\": \"B. x³ + C\", \"c\": \"C. 6x\", \"d\": \"D. x² + C\"}}","_debug_options_count":4},{"id":8316,"question":"L'intégrale ∫₀¹ x² dx représente :","option_a":"A. La longueur de la courbe y=x² entre 0 et 1","option_b":"B. L'aire sous la courbe y=x² entre 0 et 1","option_c":"C. Le volume du solide engendré par la rotation de y=x² autour de l'axe des x","option_d":"D. La dérivée de x² entre 0 et 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'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\": \"b\", \"options\": {\"a\": \"A. La longueur de la courbe y=x² entre 0 et 1\", \"b\": \"B. L'aire ","_debug_options_count":4},{"id":8317,"question":"Vrai ou Faux ? La fonction F(x) = ln(x) est une primitive de f(x) = 1\/x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(x) est bien 1\/x, donc F(x) = ln(x) est une primitive de f(x) = 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":8318,"question":"Quelle méthode utiliser pour calculer ∫x·eˣ dx ?","option_a":"A. Changement de variable","option_b":"B. Intégration par parties","option_c":"C. Décomposition en éléments simples","option_d":"D. Formule de Taylor","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégration par parties est adaptée ici car le produit x·eˣ peut être décomposé en u'·v avec u'=eˣ et v=x.","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. Changement de variable\", \"b\": \"B. Intégration par parties\", \"","_debug_options_count":4},{"id":8319,"question":"Vrai ou Faux ? ∫₀²π sin(x) dx = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de sin(x) entre 0 et 2π est nulle car les aires positives et négatives se compensent.","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":8320,"question":"Quelle est la valeur de ∫₁ᵉ (1\/x) dx ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. e","option_d":"D. ln(e)","option_e":"","option_f":"","bonne_reponse":"d","explication":"La primitive de 1\/x est ln(x). Donc ∫₁ᵉ (1\/x) dx = ln(e) - ln(1) = 1 - 0 = 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. e\", \"d\": \"D. ln(e)\"}}","_debug_options_count":4},{"id":8321,"question":"Pour calculer ∫x²·ln(x) dx, quelle méthode est la plus adaptée ?","option_a":"A. Décomposition en éléments simples","option_b":"B. Intégration par parties","option_c":"C. Changement de variable","option_d":"D. Formule de Taylor","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégration par parties est idéale ici car ln(x) est facile à dériver et x² à intégrer.","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. Décomposition en éléments simples\", \"b\": \"B. Intégration p","_debug_options_count":4},{"id":8322,"question":"Vrai ou Faux ? L'intégrale ∫₀¹ eˣ dx = e.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La primitive de eˣ est eˣ. Donc ∫₀¹ eˣ dx = e¹ - e⁰ = e - 1 ≠ e.","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":8323,"question":"Quelle est la valeur de ∫₀ᵖⁱ\/² sin(x) dx ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. -1","option_d":"D. π\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"La primitive de sin(x) est -cos(x). Donc ∫₀ᵖⁱ\/² sin(x) dx = -cos(π\/2) - (-cos(0)) = 0 + 1 = 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. -1\", \"d\": \"D. π\/2\"}}","_debug_options_count":4},{"id":8324,"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 : 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}]