Question Number 170262 by alcohol last updated on 19/May/22
$${I}_{{n}} \:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{n}} \sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:{dx} \\ $$$${show}\:{that}\:\left({n}+\mathrm{1}\right){I}_{{n}} \:=\:\left({n}+\mathrm{4}\right){I}_{{n}+\mathrm{2}} \\ $$