Задача №1445

\[ \int\frac{x^5dx}{\sqrt{a^3-x^3}} =\int\frac{x^3\cdot{x^2}dx}{\sqrt{a^3-x^3}} =\left[\begin{aligned}& u=\sqrt{a^3-x^3};\;x^3=a^3-u^2.\\& 3x^2dx=-2udu;\;x^2dx=-\frac{2udu}{3}.\end{aligned}\right]=\\ =-\frac{2}{3}\cdot\int\frac{u\cdot\left(a^3-u^2\right)du}{u} =-\frac{2}{3}\cdot\int\left(a^3-u^3\right)du =-\frac{2}{3}\cdot\left(a^3u-\frac{u^3}{3}\right)+C=\\ =-\frac{2u}{9}\cdot\left(3a^3-u^2\right)+C =-\frac{2\sqrt{a^3-x^3}}{9}\cdot\left(3a^3-\left(a^3-x^3\right)\right)+C =-\frac{2\sqrt{a^3-x^3}}{9}\cdot\left(x^3+2a^3\right)+C. \]