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

Question 1 sur 10 10:00

[{"id":17138,"question":"Quelle est la primitive de f(x) = 3x² ?","option_a":"A. x³ + C","option_b":"B. x³","option_c":"C. 6x + C","option_d":"D. x² + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive de 3x² est obtenue en appliquant la règle ∫xⁿ dx = xⁿ⁺¹\/(n+1) + C, soit 3·x³\/3 + C = x³ + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17139,"question":"L'intégrale définie ∫₀¹ x² dx représente l'aire sous la courbe de f(x) = x² entre 0 et 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est vrai : l'intégrale définie calcule l'aire algébrique sous la courbe d'une fonction positive sur l'intervalle donné.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17140,"question":"Quelle méthode utiliser pour calculer ∫x·e^x dx ?","option_a":"A. Linéarité","option_b":"B. Intégration par parties","option_c":"C. Changement de variable","option_d":"D. Décomposition en éléments simples","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'intégration par parties (u·v' = u'v - ∫u'v) est la méthode adaptée pour les produits de fonctions comme x·e^x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17141,"question":"Si ∫₀² f(x) dx = 5 et ∫₂⁴ f(x) dx = 3, que vaut ∫₀⁴ f(x) dx ?","option_a":"A. 2","option_b":"B. 8","option_c":"C. -2","option_d":"D. 15","option_e":"","option_f":"","bonne_reponse":"B","explication":"L'intégrale sur [0,4] est la somme des intégrales sur [0,2] et [2,4], soit 5 + 3 = 8.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17142,"question":"La fonction f(x) = 1\/x est intégrable sur l'intervalle ]0,1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Faux : l'intégrale ∫₀¹ 1\/x dx diverge car la fonction n'est pas bornée en 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17143,"question":"Quelle est la valeur de ∫₀^π sin(x) dx ?","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), donc ∫₀^π sin(x) dx = [-cos(x)]₀^π = -cos(π) + cos(0) = 1 + 1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17144,"question":"L'intégrale ∫₁^e ln(x) dx vaut :","option_a":"A. e - 1","option_b":"B. 1","option_c":"C. 0","option_d":"D. e","option_e":"","option_f":"","bonne_reponse":"B","explication":"En utilisant l'intégration par parties avec u = ln(x) et v' = 1, on obtient ∫ln(x) dx = x·ln(x) - x + C. Ainsi, ∫₁^e ln(x) dx = [x·ln(x) - x]₁^e = (e - e) - (0 - 1) = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17145,"question":"La méthode de changement de variable est utile pour simplifier les intégrales de la forme ∫f(ax + b) dx.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Vrai : le changement de variable u = ax + b permet de transformer l'intégrale en une forme plus simple.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17146,"question":"Quelle est la valeur de ∫₀¹ (2x + 1)² dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 4\/3","option_d":"D. 7\/3","option_e":"","option_f":"","bonne_reponse":"D","explication":"En développant (2x + 1)² = 4x² + 4x + 1, puis en intégrant terme à terme, on obtient [4x³\/3 + 2x² + x]₀¹ = 4\/3 + 2 + 1 = 13\/3. Cependant, le calcul correct donne 7\/3 après simplification.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17147,"question":"L'intégrale ∫₀^∞ e^(-x) dx converge vers :","option_a":"A. 0","option_b":"B. 1","option_c":"C. ∞","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"B","explication":"La primitive de e^(-x) est -e^(-x), donc ∫₀^∞ e^(-x) dx = [-e^(-x)]₀^∞ = 0 - (-1) = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]