Question Number 75661 by peter frank last updated on 15/Dec/19
Answered by Kunal12588 last updated on 15/Dec/19
$${U}_{{n}} =\int{x}^{{n}} \:{cos}\:{x}\:{dx} \\ $$$${u}\:=\:{x}^{{n}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\&\:\:\:\:{dv}\:=\:{cos}\:{x}\:{dx} \\ $$$$\Rightarrow{du}\:=\:{nx}^{{n}−\mathrm{1}} \:\:\:\&\:\:\:\:\:\:\:{v}\:=\:{sin}\:{x} \\ $$$$\therefore\:\:{U}_{{n}} ={x}^{{n}} \:{sin}\:{x}\:−\:{n}\int{x}^{{n}−\mathrm{1}} {sin}\:{x}\:{dx} \\ $$$$\Rightarrow{U}_{{n}} ={x}^{{n}} \:{sin}\:{x}\:−\:{nV}_{{n}−\mathrm{1}} \\ $$$${V}_{{n}} =\int{x}^{{n}} \:{sin}\:{x}\:{dx} \\ $$$${u}\:=\:{x}^{{n}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\&\:\:\:{dv}\:=\:{sin}\:{x}\:{dx} \\ $$$$\Rightarrow{du}\:=\:{nx}^{{n}−\mathrm{1}} \:\:\:\&\:\:\:\:\:\:{v}\:=\:−{cos}\:{x}\:{dx} \\ $$$$\therefore\:\:{V}_{{n}} \:=−\:{x}^{{n}} \:{cos}\:{x}\:+\:{n}\int{x}^{{n}−\mathrm{1}} \:{cos}\:{x}\:{dx} \\ $$$$\Rightarrow{V}_{{n}} \:=\:−\:{x}^{{n}} \:{cos}\:{x}\:+\:{nU}_{{n}−\mathrm{1}} \\ $$$${using}\:{red}\:{lines} \\ $$$$\therefore\:{U}_{{n}} =\:{x}^{{n}} {sin}\:{x}\:−\:{n}\left[−{x}^{{n}−\mathrm{1}} {cos}\:{x}+\left({n}−\mathrm{1}\right){U}_{{n}−\mathrm{2}} \right] \\ $$$$\Rightarrow\:{U}_{{n}} =\:{x}^{{n}} {sin}\:{x}\:+\:{nx}^{{n}−\mathrm{1}} {cos}\:{x}−{n}\left({n}−\mathrm{1}\right){U}_{{n}−\mathrm{2}} \\ $$$$\Rightarrow\:{U}_{{n}} =\:{x}^{{n}−\mathrm{1}} \left({xsin}\:{x}\:+\:{ncos}\:{x}\right)−{n}\left({n}−\mathrm{1}\right){U}_{{n}−\mathrm{2}} \\ $$$${these}\:{are}\:{just}\:{simple}\:{reduction}\:{formulae} \\ $$
Commented by peter frank last updated on 15/Dec/19
$${thank}\:{you} \\ $$