Quiz — 1578402030_6-methodes-dintegration-expliciter-une-primitive.pdf

Question 1 sur 10 20:00

[{"id":132567,"question":"Quelle méthode utiliser pour calculer ∫(2x + 1)e^(x² + x) dx ?","option_a":"Intégration directe","option_b":"Substitution","option_c":"Intégration par parties","option_d":"Décomposition en éléments simples","option_e":"","option_f":"","bonne_reponse":"b","explication":"La substitution u = x² + x simplifie l'intégrale en ∫e^u du, qui est directe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Intégration directe\", \"b\": \"Substitution\", \"c\": \"Intégration pa","_debug_options_count":4},{"id":132568,"question":"L'intégrale ∫x ln(x) dx peut se calculer par intégration par parties.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : en posant u = ln(x) et dv = x dx, on obtient une intégrale plus simple.","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":132569,"question":"Quelle est la primitive de f(x) = 3x² + 2x - 5 ?","option_a":"x³ + x² - 5x + C","option_b":"x³ + x² - 5","option_c":"3x³ + x² - 5x + C","option_d":"x³ + 2x² - 5x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 3x² est x³, celle de 2x est x², et celle de -5 est -5x. La constante C est ajoutée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x³ + x² - 5x + C\", \"b\": \"x³ + x² - 5\", \"c\": \"3x³ + x² - 5x ","_debug_options_count":4},{"id":132570,"question":"Pour calculer ∫(sin(x)cos(x)) dx, quelle substitution est adaptée ?","option_a":"u = sin(x)","option_b":"u = cos(x)","option_c":"u = sin(x)cos(x)","option_d":"u = tan(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La substitution u = sin(x) donne du = cos(x) dx, transformant l'intégrale en ∫u du = u²\/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\": \"u = sin(x)\", \"b\": \"u = cos(x)\", \"c\": \"u = sin(x)cos(x)\", \"d\": \"u ","_debug_options_count":4},{"id":132571,"question":"L'intégrale ∫(1\/(x² + 1)) dx est égale à arctan(x) + C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : la dérivée de arctan(x) est 1\/(x² + 1), donc sa primitive 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":132572,"question":"Quelle méthode est adaptée pour calculer ∫x e^(-x) dx ?","option_a":"Substitution","option_b":"Intégration par parties","option_c":"Décomposition en éléments simples","option_d":"Intégration directe","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégration par parties est idéale pour les produits de fonctions (ici, 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\": \"b\", \"options\": {\"a\": \"Substitution\", \"b\": \"Intégration par parties\", \"c\": \"Décomposit","_debug_options_count":4},{"id":132573,"question":"La primitive de f(x) = 1\/(x² - 1) est ln|(x-1)\/(x+1)| + C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux : la primitive correcte est (1\/2)ln|(x-1)\/(x+1)| + C, car la décomposition donne 1\/(x² - 1) = 1\/(2(x-1)) - 1\/(2(x+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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":132574,"question":"Quelle est la primitive de f(x) = cos(2x) ?","option_a":"sin(2x) + C","option_b":"(1\/2)sin(2x) + C","option_c":"2sin(2x) + C","option_d":"-sin(2x) + C","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de sin(2x) est 2cos(2x), donc la primitive de cos(2x) est (1\/2)sin(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\": \"sin(2x) + C\", \"b\": \"(1\/2)sin(2x) + C\", \"c\": \"2sin(2x) + C\", \"d\": ","_debug_options_count":4},{"id":132575,"question":"Pour calculer ∫(x² e^x) dx, quelle méthode est la plus efficace ?","option_a":"Substitution","option_b":"Intégration par parties","option_c":"Décomposition en éléments simples","option_d":"Intégration directe","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégration par parties est adaptée pour les produits de polynômes et d'exponentielles (ici, x² et e^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\": \"Substitution\", \"b\": \"Intégration par parties\", \"c\": \"Décomposit","_debug_options_count":4},{"id":132576,"question":"L'intégrale ∫(1\/x) dx est égale à ln|x| + C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : la dérivée de ln|x| est 1\/x, donc sa primitive est ln|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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]