Quiz — 65a14d5178b73_Enoncé-Série n° 24-Intégrales.pdf

Question 1 sur 10 20:00

[{"id":28680,"question":"Quelle est la primitive de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"A. x³ + x² - 5x + C","option_b":"B. x³ + x² - 5x","option_c":"C. 6x + 2 + C","option_d":"D. x³ + x + 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 d'intégration 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\": \"A. x³ + x² - 5x + C\", \"b\": \"B. x³ + x² - 5x\", \"c\": \"C. 6x + 2","_debug_options_count":4},{"id":28681,"question":"L'intégrale définie ∫[0,1] (2x + 1) dx est égale à :","option_a":"A. 2","option_b":"B. 1","option_c":"C. 0","option_d":"D. 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Calculons : [x² + x] de 0 à 1 = (1 + 1) - (0 + 0) = 2.","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. 2\", \"b\": \"B. 1\", \"c\": \"C. 0\", \"d\": \"D. 3\"}}","_debug_options_count":4},{"id":28682,"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. Elle est utile pour intégrer des produits de fonctions, comme uv' où u et v' sont des fonctions dérivables.","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":28683,"question":"Quelle est l'intégrale indéfinie de f(x) = e^(2x) ?","option_a":"A. (1\/2)e^(2x) + C","option_b":"B. e^(2x) + C","option_c":"C. 2e^(2x) + C","option_d":"D. (1\/2)e^x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de e^(2x) est (1\/2)e^(2x) + C, car la dérivée de (1\/2)e^(2x) est e^(2x).","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. (1\/2)e^(2x) + C\", \"b\": \"B. e^(2x) + C\", \"c\": \"C. 2e^(2x) + C\",","_debug_options_count":4},{"id":28684,"question":"L'intégrale impropre ∫[1,∞] (1\/x²) dx converge-t-elle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Cette intégrale impropre converge car lim(x→∞) [-1\/x] de 1 à ∞ = 1, qui est finie.","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":28685,"question":"Quelle est la formule de l'intégration par changement de variable pour ∫ f(g(x))g'(x) dx ?","option_a":"A. ∫ f(u) du","option_b":"B. ∫ f(u) g'(x) du","option_c":"C. ∫ f(u) g(u) du","option_d":"D. ∫ f'(u) du","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule est ∫ f(g(x))g'(x) dx = ∫ f(u) du avec u = g(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. ∫ f(u) du\", \"b\": \"B. ∫ f(u) g'(x) du\", \"c\": \"C. ∫ f(u) g","_debug_options_count":4},{"id":28686,"question":"L'aire sous la courbe de f(x) = x² entre 0 et 2 est égale à :","option_a":"A. 8\/3","option_b":"B. 4","option_c":"C. 2","option_d":"D. 16\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Calculons ∫[0,2] x² dx = [x³\/3] de 0 à 2 = 8\/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\": \"A. 8\/3\", \"b\": \"B. 4\", \"c\": \"C. 2\", \"d\": \"D. 16\/3\"}}","_debug_options_count":4},{"id":28687,"question":"La méthode d'intégration des fractions rationnelles s'applique uniquement aux fonctions polynomiales.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Elle s'applique aussi aux fractions rationnelles (P(x)\/Q(x)) où P et Q sont des polynômes.","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":28688,"question":"Quelle est l'intégrale indéfinie de f(x) = sin(3x) ?","option_a":"A. -cos(3x)\/3 + C","option_b":"B. cos(3x) + C","option_c":"C. -cos(3x) + C","option_d":"D. sin(3x)\/3 + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de sin(3x) est -cos(3x)\/3 + C, car la dérivée de -cos(3x)\/3 est sin(3x).","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. -cos(3x)\/3 + C\", \"b\": \"B. cos(3x) + C\", \"c\": \"C. -cos(3x) + C\"","_debug_options_count":4},{"id":28689,"question":"L'intégrale définie ∫[-1,1] x³ dx est égale à :","option_a":"A. 0","option_b":"B. 2","option_c":"C. -2","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction x³ est impaire, donc ∫[-1,1] x³ dx = 0.","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. 2\", \"c\": \"C. -2\", \"d\": \"D. 1\"}}","_debug_options_count":4}]