Quiz — cours_primitives.pdf

Question 1 sur 10 20:00

[{"id":65066,"question":"Quelle est la primitive de la fonction f(x) = 3x² ?","option_a":"A. x³ + C","option_b":"B. x³","option_c":"C. 6x + C","option_d":"D. x² + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de xⁿ est x^(n+1)\/(n+1) + C. Ici, n=2, donc la primitive est 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. x³\", \"c\": \"C. 6x + C\", \"d\": \"D. x² + C\"}}","_debug_options_count":4},{"id":65067,"question":"L'intégrale ∫(2x + 1) dx est égale à x² + x + C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En dérivant x² + x + C, on obtient 2x + 1, donc 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":65068,"question":"Quelle méthode utiliser pour calculer ∫x * e^x dx ?","option_a":"A. Substitution simple","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":"L'intégration par parties (u = x, dv = e^x dx) est la méthode adaptée pour les produits de fonctions.","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. Substitution simple\", \"b\": \"B. Intégration par parties\", \"c\":","_debug_options_count":4},{"id":65069,"question":"L'intégrale définie ∫(de 0 à 1) x² dx représente :","option_a":"A. La longueur de la courbe y = x² entre 0 et 1","option_b":"B. L'aire sous la courbe y = x² entre 0 et 1","option_c":"C. Le volume d'un solide de révolution","option_d":"D. La dérivée de x²","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une intégrale définie entre a et b représente l'aire algébrique sous la courbe de la fonction entre ces deux points.","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. La longueur de la courbe y = x² entre 0 et 1\", \"b\": \"B. L'air","_debug_options_count":4},{"id":65070,"question":"La primitive de 1\/(1+x²) est arctan(x) + C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de arctan(x) est 1\/(1+x²), donc la primitive est bien 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":65071,"question":"Quelle est la primitive de sin(2x) ?","option_a":"A. -cos(2x) + C","option_b":"B. -2cos(2x) + C","option_c":"C. cos(2x) + C","option_d":"D. -1\/2 cos(2x) + C","option_e":"","option_f":"","bonne_reponse":"d","explication":"En utilisant la formule ∫sin(ax) dx = -1\/a cos(ax) + C, on obtient -1\/2 cos(2x) + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A. -cos(2x) + C\", \"b\": \"B. -2cos(2x) + C\", \"c\": \"C. cos(2x) + C\",","_debug_options_count":4},{"id":65072,"question":"L'intégrale ∫(de 0 à π) sin(x) dx est égale à :","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). 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. π\"}}","_debug_options_count":4},{"id":65073,"question":"La méthode de substitution est utile pour intégrer des fonctions de la forme f(g(x)) * g'(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est exactement le principe de la substitution : poser u = g(x), alors du = g'(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":65074,"question":"Quelle est la primitive de e^(3x) ?","option_a":"A. e^(3x) + C","option_b":"B. 1\/3 e^(3x) + C","option_c":"C. 3e^(3x) + C","option_d":"D. e^(3x+1) + C","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la formule ∫e^(ax) dx = 1\/a e^(ax) + C, on obtient 1\/3 e^(3x) + 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^(3x) + C\", \"b\": \"B. 1\/3 e^(3x) + C\", \"c\": \"C. 3e^(3x) + C\", ","_debug_options_count":4},{"id":65075,"question":"L'intégrale ∫(de -1 à 1) x³ dx est égale à :","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction x³ est impaire, donc l'intégrale entre -1 et 1 est nulle (aires opposées).","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. 2\", \"d\": \"D. -1\"}}","_debug_options_count":4}]