Задача 71948 integrate x ^ 2 * 1/(root(7 - x ^ 3, 6))...

6446f0f26e916e3d0082e6e7

06.06.2023 12:07:59

integrate x ^ 2 * 1/(root(7 - x ^ 3, 6)) dx

626fab9a354ab65fb2f43abb

06.06.2023 13:59:04

★

[m]\int \frac{x^2}{\sqrt[6]{7-x^3}}dx[/m]
Замена 7 - x^3 = y; dy = -3x^2 dx; x^2 dx = -1/3*dy
[m]-\frac{1}{3} \int \frac{dy}{\sqrt[6]{y}} = -\frac{1}{3} \int y^{-1/6}dy = -\frac{1}{3} \frac{y^{-1/6+1}}{-1/6+1} + C = -\frac{1}{3} \cdot \frac{6}{5} y^{5/6}+C=[/m]
[m]-\frac{2}{5} (7-x^3)^{5/6}+C[/m]