Quiz — serie primitive.pdf

Question 1 sur 10 20:00

[{"id":30950,"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 + 3x + C","option_d":"D. 3x³ + 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 de 3x² est 3*(x³\/3) + C = 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 + 3x + C\", \"d\": \"D. 3","_debug_options_count":4},{"id":30951,"question":"L'intégrale de 0 à 1 de la fonction f(x) = 2x est égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 2x est x² + C. L'intégrale de 0 à 1 est [x²] de 0 à 1 = 1 - 0 = 1. L'affirmation est donc 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":30952,"question":"Quelle méthode utiliser pour calculer la primitive de f(x) = x*e^x ?","option_a":"A. Intégration par parties","option_b":"B. Changement de variable","option_c":"C. Décomposition en éléments simples","option_d":"D. Formule de Taylor","option_e":"","option_f":"","bonne_reponse":"a","explication":"La méthode la plus adaptée est l'intégration par parties, car la fonction est un produit de x et e^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\": \"A. Intégration par parties\", \"b\": \"B. Changement de variable\", \"","_debug_options_count":4},{"id":30953,"question":"L'intégrale de -1 à 1 de la fonction f(x) = x³ est égale à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de x³ est x⁴\/4 + C. L'intégrale de -1 à 1 est [x⁴\/4] de -1 à 1 = (1\/4) - (1\/4) = 0. 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":30954,"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. e^x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de arctan(x) est 1\/(1+x²). Donc, la primitive de 1\/(1+x²) est arctan(x) + 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. arctan(x) + C\", \"b\": \"B. ln(1+x²) + C\", \"c\": \"C. x + C\", \"d\":","_debug_options_count":4},{"id":30955,"question":"L'intégrale de 0 à π de la fonction f(x) = sin(x) est égale à 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de sin(x) est -cos(x) + C. L'intégrale de 0 à π est [-cos(x)] de 0 à π = -(-1) - (-1) = 2. 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":30956,"question":"Quelle est la primitive de f(x) = e^(2x) ?","option_a":"A. e^(2x) + C","option_b":"B. (1\/2)*e^(2x) + C","option_c":"C. 2*e^(2x) + C","option_d":"D. e^x + C","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de e^(2x) est 2*e^(2x). Donc, la primitive de e^(2x) est (1\/2)*e^(2x) + 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. e^(2x) + C\", \"b\": \"B. (1\/2)*e^(2x) + C\", \"c\": \"C. 2*e^(2x) + C","_debug_options_count":4},{"id":30957,"question":"L'intégrale de 0 à 1 de la fonction f(x) = 1\/x est définie.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction 1\/x n'est pas définie en x=0, donc son intégrale sur [0,1] n'est pas définie. L'affirmation est fausse.","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":30958,"question":"Quelle est la primitive de f(x) = ln(x) ?","option_a":"A. x*ln(x) - x + C","option_b":"B. x + C","option_c":"C. ln(x)\/x + C","option_d":"D. e^x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de ln(x) est x*ln(x) - x + C, obtenue par 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\": \"A. x*ln(x) - x + C\", \"b\": \"B. x + C\", \"c\": \"C. ln(x)\/x + C\", \"d\":","_debug_options_count":4},{"id":30959,"question":"L'intégrale de 0 à 2π de la fonction f(x) = cos(x) est égale à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de cos(x) est sin(x) + C. L'intégrale de 0 à 2π est [sin(x)] de 0 à 2π = 0 - 0 = 0. 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}]