Quiz — Serie 24- Intégrales.pdf

Question 1 sur 10 20:00

[{"id":9923,"question":"Quelle est la primitive de la fonction f(x) = 2x ?","option_a":"x² + C","option_b":"x²","option_c":"2x² + C","option_d":"x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 2x est x² + C, car la dérivée de x² est 2x. C représente la constante d’intégration.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x² + C\", \"b\": \"x²\", \"c\": \"2x² + C\", \"d\": \"x + C\"}}","_debug_options_count":4},{"id":9924,"question":"L’intégrale définie ∫(de 0 à 1) x² dx est égale à :","option_a":"1\/3","option_b":"1","option_c":"0","option_d":"2\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"L’intégrale de x² entre 0 et 1 est [x³\/3] de 0 à 1 = 1\/3 - 0 = 1\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"1\/3\", \"b\": \"1\", \"c\": \"0\", \"d\": \"2\/3\"}}","_debug_options_count":4},{"id":9925,"question":"La méthode d’intégration par parties s’applique à :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. La méthode par parties est utile pour intégrer des produits de fonctions, comme ∫u(x)v'(x)dx = u(x)v(x) - ∫u'(x)v(x)dx.","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":9926,"question":"Quelle est la valeur de ∫(de 0 à π) sin(x) dx ?","option_a":"0","option_b":"2","option_c":"-2","option_d":"π","option_e":"","option_f":"","bonne_reponse":"b","explication":"L’intégrale de sin(x) entre 0 et π est [-cos(x)] de 0 à π = -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\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"2\", \"c\": \"-2\", \"d\": \"π\"}}","_debug_options_count":4},{"id":9927,"question":"L’intégrale ∫(de 1 à e) (1\/x) dx est égale à :","option_a":"0","option_b":"1","option_c":"e","option_d":"ln(e)","option_e":"","option_f":"","bonne_reponse":"b","explication":"L’intégrale de 1\/x est ln|x|, donc ∫(de 1 à e) (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\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"e\", \"d\": \"ln(e)\"}}","_debug_options_count":4},{"id":9928,"question":"Une intégrale impropre est une intégrale où :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Une intégrale impropre est une intégrale où l’une des bornes est infinie ou la fonction n’est pas définie en un point de l’intervalle d’intégration.","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":9929,"question":"Quelle méthode utiliser pour calculer ∫x e^x dx ?","option_a":"Substitution","option_b":"Par parties","option_c":"Décomposition en éléments simples","option_d":"Intégration directe","option_e":"","option_f":"","bonne_reponse":"b","explication":"La méthode par parties est adaptée pour intégrer x e^x, car elle permet de réduire le degré de 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\": \"Par parties\", \"c\": \"Décomposition en élém","_debug_options_count":4},{"id":9930,"question":"L’intégrale ∫(de -1 à 1) x³ dx est égale à :","option_a":"0","option_b":"2","option_c":"-2","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction x³ est impaire, donc son intégrale sur un intervalle symétrique autour de 0 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\": \"0\", \"b\": \"2\", \"c\": \"-2\", \"d\": \"1\"}}","_debug_options_count":4},{"id":9931,"question":"La formule de Taylor permet d’approximer une fonction par un polynôme. Vrai ou faux ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. La formule de Taylor exprime une fonction comme une somme infinie de termes polynomiaux autour d’un point.","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":9932,"question":"Quelle est la valeur de ∫(de 0 à 1) (3x² + 2x + 1) dx ?","option_a":"2","option_b":"3","option_c":"1","option_d":"4","option_e":"","option_f":"","bonne_reponse":"b","explication":"L’intégrale de 3x² + 2x + 1 est x³ + x² + x, donc [x³ + x² + x] de 0 à 1 = 1 + 1 + 1 = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"3\", \"c\": \"1\", \"d\": \"4\"}}","_debug_options_count":4}]