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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

Un=∫xncosxdxu=xn&dv=cosxdx⇒du=nxn−1&v=sinx∴Un=xnsinx−n∫xn−1sinxdx⇒Un=xnsinx−nVn−1Vn=∫xnsinxdxu=xn&dv=sinxdx⇒du=nxn−1&v=−cosxdx∴Vn=−xncosx+n∫xn−1cosxdx⇒Vn=−xncosx+nUn−1usingredlines∴Un=xnsinx−n[−xn−1cosx+(n−1)Un−2]⇒Un=xnsinx+nxn−1cosx−n(n−1)Un−2⇒Un=xn−1(xsinx+ncosx)−n(n−1)Un−2thesearejustsimplereductionformulae

Commented by peter frank last updated on 15/Dec/19

thank you

thankyou

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