Задача №1453

\[ \int\frac{\sqrt{\left(9-x^2\right)^3}}{x^6}dx =\left[\begin{aligned}& x=3\sin{t}.\\& dx=3\cos{t}dt.\end{aligned}\right] =\int\frac{27\cos^3{t}\cdot{3}\cos{t}dt}{729\sin^6{t}} =\frac{1}{9}\int\frac{\cos^4{t}}{\sin^6{t}}dt =-\frac{1}{9}\int\ctg^4{t}\,d(\ctg{t})=\\ =-\frac{\ctg^5{t}}{45}+C =-\frac{\cos^5{t}}{45\sin^5{t}}+C =-\frac{\sqrt{\left(1-\sin^2{t}\right)^5}}{45\sin^5{t}}+C =-\frac{\sqrt{\left(1-\frac{x^2}{9}\right)^5}}{45\cdot\left(\frac{x}{3}\right)^5}+C =-\frac{\sqrt{\left(9-x^2\right)^5}}{45x^5}+C \]