Question Number 220948 by fantastic last updated on 21/May/25
$$\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{5}−{x}^{\mathrm{6}} }{dx} \\ $$
Answered by SdC355 last updated on 21/May/25
$${x}^{\mathrm{3}} ={u} \\ $$$$\frac{\mathrm{d}{u}}{\mathrm{d}{x}}=\mathrm{3}{x}^{\mathrm{2}} \:\rightarrow\:\mathrm{d}{u}=\mathrm{3}{x}^{\mathrm{2}} \mathrm{d}{x} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}\int\:\:\sqrt{\mathrm{5}−{u}^{\mathrm{2}} }\:\mathrm{d}{u}=\frac{\mathrm{1}}{\mathrm{6}}\left({x}^{\mathrm{3}} \sqrt{\mathrm{5}−{x}^{\mathrm{6}} }+\mathrm{5}\centerdot\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}{x}^{\mathrm{3}} \right)+\mathrm{Const}\right. \\ $$$$ \\ $$$$ \\ $$