Quiz — coursIntegQlq.pdf

Question 1 sur 10 20:00

[{"id":53257,"question":"Quelle est la primitive de la fonction f(x) = 3x² ?","option_a":"x³ + C","option_b":"x³","option_c":"3x³ + C","option_d":"6x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de x^n est x^(n+1)\/(n+1). Ici, n=2, donc la primitive est x³\/3. Multiplié par 3, cela donne 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\": \"x³ + C\", \"b\": \"x³\", \"c\": \"3x³ + C\", \"d\": \"6x + C\"}}","_debug_options_count":4},{"id":53258,"question":"L'intégrale de 0 à π de sin(x) dx est égale à :","option_a":"0","option_b":"1","option_c":"2","option_d":"π","option_e":"","option_f":"","bonne_reponse":"c","explication":"La primitive de sin(x) est -cos(x). Évaluée entre 0 et π, on obtient -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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"π\"}}","_debug_options_count":4},{"id":53259,"question":"Vrai ou Faux ? L'intégrale d'une fonction paire sur [-a, a] est toujours nulle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. L'intégrale d'une fonction paire sur [-a, a] est égale à 2 fois l'intégrale de 0 à a. Elle n'est nulle que si la fonction est nulle sur [0, a].","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":53260,"question":"Quelle méthode utiliser pour calculer ∫ x * e^x dx ?","option_a":"Substitution","option_b":"Intégration par parties","option_c":"Décomposition en éléments simples","option_d":"Aucune des méthodes ci-dessus","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégration par parties est adaptée aux produits de fonctions. Ici, on pose u = x et dv = e^x dx, donc du = dx et v = 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":53261,"question":"L'intégrale de 1\/(1+x²) dx est égale à :","option_a":"ln|1+x²| + C","option_b":"arctan(x) + C","option_c":"1\/(1+x²) + C","option_d":"x²\/2 + C","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de arctan(x) est 1\/(1+x²), donc la 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\": \"b\", \"options\": {\"a\": \"ln|1+x²| + C\", \"b\": \"arctan(x) + C\", \"c\": \"1\/(1+x²) + C\", \"d\": ","_debug_options_count":4},{"id":53262,"question":"Vrai ou Faux ? L'intégrale de f(x) = 2x sur [1, 3] représente l'aire sous la courbe de f entre 1 et 3.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. L'intégrale définie d'une fonction positive sur un intervalle représente l'aire sous la courbe entre les bornes données.","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":53263,"question":"Quelle est la valeur de ∫ (2x + 1) dx entre 0 et 2 ?","option_a":"4","option_b":"6","option_c":"8","option_d":"10","option_e":"","option_f":"","bonne_reponse":"b","explication":"La primitive de 2x+1 est x² + x. Évaluée entre 0 et 2, on obtient (4 + 2) - (0 + 0) = 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"4\", \"b\": \"6\", \"c\": \"8\", \"d\": \"10\"}}","_debug_options_count":4},{"id":53264,"question":"Vrai ou Faux ? L'intégrale de f(x) = e^(-x) sur [0, +∞[ est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. L'intégrale ∫ e^(-x) dx entre 0 et +∞ vaut 1, donc elle est convergente.","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":53265,"question":"Quelle méthode utiliser pour calculer ∫ (x² + 1)\/(x³ + x) dx ?","option_a":"Substitution","option_b":"Intégration par parties","option_c":"Décomposition en éléments simples","option_d":"Aucune des méthodes ci-dessus","option_e":"","option_f":"","bonne_reponse":"c","explication":"La décomposition en éléments simples est adaptée aux fractions rationnelles. On décompose (x² + 1)\/(x³ + x) en 1\/x + (x)\/(x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Substitution\", \"b\": \"Intégration par parties\", \"c\": \"Décomposit","_debug_options_count":4},{"id":53266,"question":"L'intégrale de f(x) = cos(2x) dx est égale à :","option_a":"sin(2x) + C","option_b":"2sin(2x) + C","option_c":"(1\/2)sin(2x) + C","option_d":"-sin(2x) + C","option_e":"","option_f":"","bonne_reponse":"c","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\": \"c\", \"options\": {\"a\": \"sin(2x) + C\", \"b\": \"2sin(2x) + C\", \"c\": \"(1\/2)sin(2x) + C\", \"d\": ","_debug_options_count":4}]