Maîtrisez les intégrales : QCM et Vrai/Faux pour la Terminale

Question 1 sur 10 10:00

[{"id":27268,"question":"Quelle est la primitive de la fonction f(x) = 3x² ?","option_a":"x³ + C","option_b":"x³","option_c":"6x + C","option_d":"x² + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive de x^n 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},{"id":27269,"question":"L'intégrale définie ∫(de 0 à 1) x dx est égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"∫x dx = x²\/2. Calculée entre 0 et 1, on obtient (1²\/2 - 0²\/2) = 0.5 ≠ 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27270,"question":"Quelle méthode utiliser pour calculer ∫x.e^x dx ?","option_a":"Changement de variable","option_b":"Intégration par parties","option_c":"Décomposition en éléments simples","option_d":"Primitive directe","option_e":"","option_f":"","bonne_reponse":"B","explication":"La méthode d'intégration par parties (u = x, dv = e^x dx) est adaptée pour ce type d'intégrale.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27271,"question":"L'aire sous la courbe de f(x) = x² entre x=0 et x=2 est égale à 8\/3.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"∫(0 à 2) x² dx = [x³\/3] (0 à 2) = 8\/3. L'affirmation est donc vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27272,"question":"Quelle est la valeur de ∫(de -1 à 1) x³ dx ?","option_a":"0","option_b":"2","option_c":"-2","option_d":"1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction x³ est impaire. L'intégrale d'une fonction impaire sur un intervalle symétrique est nulle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27273,"question":"La formule d'intégration par parties est : ∫u.v' dx = u.v - ∫u'.v dx.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La formule correcte est bien ∫u.v' dx = u.v - ∫u'.v dx (théorème de Leibniz-Newton).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27274,"question":"Quelle intégrale permet de calculer l'aire entre les courbes y = x² et y = x + 2 ?","option_a":"∫(de 0 à 2) (x + 2 - x²) dx","option_b":"∫(de -1 à 2) (x² - x - 2) dx","option_c":"∫(de -1 à 2) (x + 2 - x²) dx","option_d":"∫(de 0 à 2) (x² - x - 2) dx","option_e":"","option_f":"","bonne_reponse":"C","explication":"L'aire entre deux courbes f(x) et g(x) est donnée par ∫|f(x) - g(x)| dx sur l'intervalle où elles se croisent. Ici, f(x) = x + 2 et g(x) = x².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27275,"question":"L'intégrale ∫(de 0 à π) sin(x) dx est égale à 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"∫sin(x) dx = -cos(x). Calculée entre 0 et π, on obtient -cos(π) - (-cos(0)) = 1 - (-1) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27276,"question":"Quelle est la primitive de f(x) = 1\/(1 + x²) ?","option_a":"arctan(x) + C","option_b":"ln(1 + x²) + C","option_c":"arcsin(x) + C","option_d":"x\/(1 + x²) + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de arctan(x) est 1\/(1 + x²), donc sa primitive est arctan(x) + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27277,"question":"L'intégrale ∫(de 0 à 1) e^(-x) dx est égale à 1 - 1\/e.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"∫e^(-x) dx = -e^(-x). Calculée entre 0 et 1, on obtient -e^(-1) - (-e^(0)) = 1 - 1\/e.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]