Question Number 66332 by mathmax by abdo last updated on 12/Aug/19
$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}−{x}}{dx} \\ $$$$\left.\mathrm{1}\right){calculate}\:{A}_{\mathrm{0}} \:{and}\:{A}_{\mathrm{1}} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\forall{n}\in{N}^{\bigstar} \:\:\:\:\left(\mathrm{3}+\mathrm{2}{n}\right){A}_{{n}} =\mathrm{2}{nA}_{{n}−\mathrm{1}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{A}_{{n}} \:{interms}\:{of}\:{n}. \\ $$