Question Number 170662 by MikeH last updated on 28/May/22
$$\mathrm{Given}\:\mathrm{that}\: \\ $$$${I}_{{n}} \:=\:\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} \sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:{dx} \\ $$$$\mathrm{Show}\:\mathrm{that} \\ $$$$\:{I}_{{n}+\mathrm{2}} \:=\:\left(\frac{{I}_{{n}+\mathrm{1}} }{{I}_{{n}+\mathrm{4}} }\right){I}_{{n}} \\ $$