Задача 46094 ...
Условие
математика
572
Решение
★
[m]\int^{1}_{0}\frac{x^2dx}{\sqrt[4]{1-x^3}} =\lim_{\varepsilon\to 0}\int^{1-\varepsilon }_{0}\frac{x^2dx}{\sqrt[4]{1-x^3}} =-\frac{1}{3}\lim_{\varepsilon \to 0}\int^{1-\varepsilon }_{0}\frac{(-3x^2dx)}{\sqrt[4]{1-x^3}}=[/m][m]-\frac{1}{3}\lim_{\varepsilon \to 0}\int^{1-\varepsilon }_{0}(1-x^3)^{-\frac{1}{4}}d(1-x^3) =[/m]
[m]=-\frac{1}{3}\lim_{\varepsilon \to 0}\frac{(1-x^3)^{-\frac{1}{4}+1}}{-\frac{1}{4}+1}|^{1-\varepsilon }_{0}=-\frac{4}{9}\cdot (\lim_{\varepsilon \to 0}\sqrt[4]{(1-x^2)^3}|^{1-\varepsilon }_{0}=[/m]
[m]=-\frac{4}{9}\cdot (\lim_{\varepsilon \to 0}\sqrt[4]{(1-(1-\varepsilon)^2)^3}-1)=\frac{4}{9}[/m]
[b]Cходится.[/b]
