Quiz — cours_integrales.pdf

Question 1 sur 10 20:00

[{"id":115714,"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. 3x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 3x² est x³ + C, car la dérivée de x³ est 3x².","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. 3x + C\"}}","_debug_options_count":4},{"id":115715,"question":"L'intégrale ∫(de 0 à 1) 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":"C'est vrai. L'intégrale d'une fonction positive entre deux bornes représente l'aire sous la courbe entre ces bornes.","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":115716,"question":"Quelle méthode utilise-t-on pour calculer ∫ x·eˣ dx ?","option_a":"A. Intégration par substitution","option_b":"B. Intégration par parties","option_c":"C. Décomposition en éléments simples","option_d":"D. Changement de variable","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise l'intégration par parties, car la fonction est un produit de x et eˣ.","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 substitution\", \"b\": \"B. Intégration par part","_debug_options_count":4},{"id":115717,"question":"L'intégrale ∫(de -1 à 1) x³ dx est égale à :","option_a":"A. 0","option_b":"B. 1","option_c":"C. -1","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction x³ est impaire, donc son intégrale entre -1 et 1 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. -1\", \"d\": \"D. 2\"}}","_debug_options_count":4},{"id":115718,"question":"La formule ∫ u'(x)·v(x) dx = u(x)·v(x) - ∫ u(x)·v'(x) dx s'appelle :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai. C'est la formule d'intégration par parties.","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":115719,"question":"Quelle est la valeur de ∫(de 0 à π) sin(x) dx ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. π","option_e":"","option_f":"","bonne_reponse":"c","explication":"La primitive de sin(x) est -cos(x), donc ∫(de 0 à π) sin(x) dx = -cos(π) - (-cos(0)) = 1 - (-1) = 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. π\"}}","_debug_options_count":4},{"id":115720,"question":"L'intégrale ∫(de 1 à e) (1\/x) dx représente :","option_a":"A. L'aire sous la courbe de 1\/x entre 1 et e","option_b":"B. Le logarithme naturel de e","option_c":"C. La dérivée de ln(x)","option_d":"D. La valeur de e","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette intégrale représente l'aire sous la courbe de 1\/x entre 1 et e, et vaut ln(e) - ln(1) = 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. L'aire sous la courbe de 1\/x entre 1 et e\", \"b\": \"B. Le logari","_debug_options_count":4},{"id":115721,"question":"La dérivée de l'intégrale ∫(de a à x) f(t) dt est :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai. Par le théorème fondamental de l'analyse, la dérivée de cette intégrale est f(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":115722,"question":"Quelle est la primitive de la fonction f(x) = e²ˣ ?","option_a":"A. e²ˣ + C","option_b":"B. (1\/2)e²ˣ + C","option_c":"C. 2e²ˣ + C","option_d":"D. eˣ + C","option_e":"","option_f":"","bonne_reponse":"b","explication":"La primitive de e²ˣ est (1\/2)e²ˣ + C, car la dérivée de (1\/2)e²ˣ est e²ˣ.","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. e²ˣ + C\", \"b\": \"B. (1\/2)e²ˣ + C\", \"c\": \"C. 2e²ˣ + C\", \"d","_debug_options_count":4},{"id":115723,"question":"L'intégrale ∫(de 0 à 1) (2x + 1) dx est égale à :","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"c","explication":"La primitive de 2x + 1 est x² + x, donc ∫(de 0 à 1) (2x + 1) dx = (1 + 1) - (0 + 0) = 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\": \"A. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4}]