Quiz — Primitives par changement de variable, III.pdf

Question 1 sur 10 20:00

[{"id":21199,"question":"Quelle substitution permet de calculer la primitive de f(x) = 2x * e^(x²) ?","option_a":"A. u = x²","option_b":"B. u = e^x","option_c":"C. u = 2x","option_d":"D. u = x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La substitution u = x² permet d'obtenir du du = 2x dx, ce qui simplifie l'intégrale en ∫ e^u du.","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. u = x²\", \"b\": \"B. u = e^x\", \"c\": \"C. u = 2x\", \"d\": \"D. u = x\"","_debug_options_count":4},{"id":21200,"question":"La primitive de f(x) = 1\/(3x + 5) est ln|3x + 5| + C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai, car la dérivée de ln|3x + 5| est 3\/(3x + 5), donc la primitive est (1\/3)ln|3x + 5| + 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":21201,"question":"Quelle est la primitive de f(x) = sin(5x + 2) ?","option_a":"A. -cos(5x + 2) + C","option_b":"B. (1\/5)cos(5x + 2) + C","option_c":"C. -cos(5x + 2)\/5 + C","option_d":"D. cos(5x + 2)\/5 + C","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée de -cos(5x + 2)\/5 est sin(5x + 2), donc c'est la bonne primitive.","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. -cos(5x + 2) + C\", \"b\": \"B. (1\/5)cos(5x + 2) + C\", \"c\": \"C. -c","_debug_options_count":4},{"id":21202,"question":"Pour calculer ∫ x * √(x² + 1) dx, quelle substitution est la plus adaptée ?","option_a":"A. u = x","option_b":"B. u = x² + 1","option_c":"C. u = √x","option_d":"D. u = x²","option_e":"","option_f":"","bonne_reponse":"b","explication":"La substitution u = x² + 1 donne du = 2x dx, ce qui simplifie l'intégrale en (1\/2)∫ √u du.","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. u = x\", \"b\": \"B. u = x² + 1\", \"c\": \"C. u = √x\", \"d\": \"D. u ","_debug_options_count":4},{"id":21203,"question":"La primitive de f(x) = e^(2x) est e^(2x) + C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux, car la dérivée de e^(2x) est 2e^(2x), donc la primitive est (1\/2)e^(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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":21204,"question":"Quelle est la primitive de f(x) = 1\/(1 + x²) ?","option_a":"A. arctan(x) + C","option_b":"B. ln(1 + x²) + C","option_c":"C. arccos(x) + C","option_d":"D. tan(x) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de arctan(x) est 1\/(1 + x²), donc c'est la primitive correcte.","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. arctan(x) + C\", \"b\": \"B. ln(1 + x²) + C\", \"c\": \"C. arccos(x) ","_debug_options_count":4},{"id":21205,"question":"Pour calculer ∫ (2x + 3) * e^(x² + 3x) dx, quelle substitution utiliser ?","option_a":"A. u = x","option_b":"B. u = x² + 3x","option_c":"C. u = 2x + 3","option_d":"D. u = e^x","option_e":"","option_f":"","bonne_reponse":"b","explication":"La substitution u = x² + 3x donne du = (2x + 3) dx, ce qui simplifie l'intégrale en ∫ e^u du.","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. u = x\", \"b\": \"B. u = x² + 3x\", \"c\": \"C. u = 2x + 3\", \"d\": \"D.","_debug_options_count":4},{"id":21206,"question":"La primitive de f(x) = cos(3x) est sin(3x) + C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux, car la dérivée de sin(3x) est 3cos(3x), donc la primitive est (1\/3)sin(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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":21207,"question":"Quelle est la primitive de f(x) = 1\/√(1 - x²) ?","option_a":"A. arcsin(x) + C","option_b":"B. arccos(x) + C","option_c":"C. arctan(x) + C","option_d":"D. ln|x| + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de arcsin(x) est 1\/√(1 - x²), donc c'est la primitive correcte.","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. arcsin(x) + C\", \"b\": \"B. arccos(x) + C\", \"c\": \"C. arctan(x) + ","_debug_options_count":4},{"id":21208,"question":"Pour calculer ∫ (x² + 1) * (x³ + 3x)² dx, quelle substitution est adaptée ?","option_a":"A. u = x","option_b":"B. u = x² + 1","option_c":"C. u = x³ + 3x","option_d":"D. u = x³","option_e":"","option_f":"","bonne_reponse":"c","explication":"La substitution u = x³ + 3x donne du = (3x² + 3) dx = 3(x² + 1) dx, ce qui simplifie l'intégrale en (1\/3)∫ u² du.","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. u = x\", \"b\": \"B. u = x² + 1\", \"c\": \"C. u = x³ + 3x\", \"d\": \"D","_debug_options_count":4}]