Quiz — تمارين التكامل إعداد أضرضور مصطفى.pdf

Question 1 sur 10 20:00

[{"id":4061,"question":"Quelle est la primitive de la fonction f(x) = 3x² + 2x - 1 ?","option_a":"A. x³ + x² - x + C","option_b":"B. x³ + x² - x","option_c":"C. 6x + 2 + C","option_d":"D. x³ + x² - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 3x² est x³, celle de 2x est x², et celle de -1 est -x. On ajoute la constante C pour une intégrale indéfinie.","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 - x + C\", \"b\": \"B. x³ + x² - x\", \"c\": \"C. 6x + 2 +","_debug_options_count":4},{"id":4062,"question":"L'intégrale ∫(sin(x) + cos(x)) dx est égale à :","option_a":"A. -cos(x) + sin(x) + C","option_b":"B. cos(x) + sin(x) + C","option_c":"C. -cos(x) - sin(x) + C","option_d":"D. cos(x) - sin(x) + C","option_e":"","option_f":"","bonne_reponse":"b","explication":"La primitive de sin(x) est -cos(x) et celle de cos(x) est sin(x). La somme donne 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\": \"b\", \"options\": {\"a\": \"A. -cos(x) + sin(x) + C\", \"b\": \"B. cos(x) + sin(x) + C\", \"c\": \"C.","_debug_options_count":4},{"id":4063,"question":"L'intégration par parties est utile pour intégrer des produits de fonctions.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégration par parties est une technique spécifique pour les produits de fonctions, basée sur la formule ∫u dv = uv - ∫v 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":4064,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (2x + 1) dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"c","explication":"La primitive de 2x + 1 est x² + x. En évaluant entre 0 et 1, on obtient (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\": \"c\", \"options\": {\"a\": \"A. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":4065,"question":"L'intégrale ∫(1\/x) dx est égale à :","option_a":"A. ln|x| + C","option_b":"B. 1\/x² + C","option_c":"C. x ln|x| + C","option_d":"D. e^x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 1\/x est ln|x| + C, une formule fondamentale à retenir.","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. ln|x| + C\", \"b\": \"B. 1\/x² + C\", \"c\": \"C. x ln|x| + C\", \"d\": \"","_debug_options_count":4},{"id":4066,"question":"Une intégrale définie représente toujours une aire positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une intégrale définie peut être négative si la fonction est négative sur l'intervalle considéré. L'aire est toujours positive, mais l'intégrale peut être négative.","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":4067,"question":"Quelle technique utiliser pour calculer ∫x e^x dx ?","option_a":"A. Intégration par parties","option_b":"B. Changement de variable","option_c":"C. Décomposition en éléments simples","option_d":"D. Formule de Taylor","option_e":"","option_f":"","bonne_reponse":"a","explication":"Ce produit de x et e^x nécessite l'intégration par parties, avec u = x et dv = e^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\": \"A. Intégration par parties\", \"b\": \"B. Changement de variable\", \"","_debug_options_count":4},{"id":4068,"question":"L'intégrale ∫₀^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). En évaluant 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":4069,"question":"La méthode de changement de variable est utile pour simplifier les intégrales.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le changement de variable permet de transformer une intégrale complexe en une intégrale plus simple, en posant 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":4070,"question":"Quelle est la primitive de f(x) = e^(2x) ?","option_a":"A. e^(2x) + C","option_b":"B. (1\/2) e^(2x) + C","option_c":"C. 2 e^(2x) + C","option_d":"D. e^x + C","option_e":"","option_f":"","bonne_reponse":"b","explication":"La primitive de e^(kx) est (1\/k) e^(kx) + C. Ici, k = 2, 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\": \"A. e^(2x) + C\", \"b\": \"B. (1\/2) e^(2x) + C\", \"c\": \"C. 2 e^(2x) + C","_debug_options_count":4}]