Quiz — Intégrales de fonctions trigonométriques.pdf

Question 1 sur 10 20:00

[{"id":39040,"question":"Quelle est la primitive de sin(3x) ?","option_a":"-cos(3x)\/3 + C","option_b":"cos(3x)\/3 + C","option_c":"-3cos(3x) + C","option_d":"sin(3x)\/3 + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de sin(ax) est -cos(ax)\/a + C. Ici, a = 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\": \"-cos(3x)\/3 + C\", \"b\": \"cos(3x)\/3 + C\", \"c\": \"-3cos(3x) + C\", \"d\":","_debug_options_count":4},{"id":39041,"question":"L'intégrale de cos²(x) entre 0 et π est égale à π\/2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de réduction cos²(x) = (1 + cos(2x))\/2, l'intégrale vaut π\/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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":39042,"question":"Quelle méthode utiliser pour intégrer x·sin(x) ?","option_a":"Substitution","option_b":"Intégration par parties","option_c":"Formule de réduction","option_d":"Décomposition en éléments simples","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégration par parties est adaptée pour les produits de fonctions polynomiales et trigonométriques.","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\": \"Formule de ","_debug_options_count":4},{"id":39043,"question":"L'intégrale de tan(x) entre 0 et π\/4 est égale à :","option_a":"0","option_b":"1","option_c":"ln(2)","option_d":"π\/4","option_e":"","option_f":"","bonne_reponse":"c","explication":"La primitive de tan(x) est -ln|cos(x)|. En évaluant entre 0 et π\/4, on obtient ln(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\": \"ln(2)\", \"d\": \"π\/4\"}}","_debug_options_count":4},{"id":39044,"question":"La dérivée de sin²(x) est :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de sin²(x) est 2sin(x)cos(x) = sin(2x) (formule de dérivation composée).","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":39045,"question":"Quelle est la valeur de ∫[0, π\/2] sin(x)cos(x) dx ?","option_a":"0","option_b":"1\/2","option_c":"π\/4","option_d":"1","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la substitution u = sin(x), l'intégrale devient ∫[0,1] u du = [u²\/2]₀¹ = 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\": \"1\/2\", \"c\": \"π\/4\", \"d\": \"1\"}}","_debug_options_count":4},{"id":39046,"question":"Peut-on intégrer sin(x) + cos(x) terme à terme ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, l'intégrale d'une somme est la somme des intégrales. ∫(sin(x) + cos(x)) dx = -cos(x) + sin(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":39047,"question":"L'intégrale de sec²(x) est :","option_a":"tan(x) + C","option_b":"sec(x) + C","option_c":"ln|sec(x)| + C","option_d":"sin(x) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de tan(x) est sec²(x), donc sa primitive est tan(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\": \"tan(x) + C\", \"b\": \"sec(x) + C\", \"c\": \"ln|sec(x)| + C\", \"d\": \"sin(","_debug_options_count":4},{"id":39048,"question":"Quelle formule permet de simplifier ∫sin²(x) dx ?","option_a":"sin²(x) = (1 - cos(2x))\/2","option_b":"sin²(x) = 1 - cos²(x)","option_c":"sin²(x) = 2sin(x)cos(x)","option_d":"sin²(x) = cos(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule de réduction sin²(x) = (1 - cos(2x))\/2 permet de simplifier l'intégrale.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"sin²(x) = (1 - cos(2x))\/2\", \"b\": \"sin²(x) = 1 - cos²(x)\", \"c\":","_debug_options_count":4},{"id":39049,"question":"L'intégrale de 1\/(1 + sin(x)) entre 0 et π\/2 est :","option_a":"0","option_b":"1","option_c":"π\/2","option_d":"2","option_e":"","option_f":"","bonne_reponse":"b","explication":"En multipliant par (1 - sin(x))\/(1 - sin(x)), on obtient une intégrale simple valant 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\": \"π\/2\", \"d\": \"2\"}}","_debug_options_count":4}]