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Question-75661




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

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