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Question Number 172996 by Mathspace last updated on 04/Jul/22

let U_n =∫_0 ^1 (√(1−x^n ))ln^2 xdx 1)lim U_n ? 2)equivalent of U_n (n→∞)

letUn=∫011−xnln2xdx1)limUn?2)equivalentofUn(n→∞)

Answered by aleks041103 last updated on 05/Jul/22

U_n =∫_0 ^1 (√(1−x^n ))ln^2 x dx⇒x∈(0,1) ⇒lim_(n→∞) x^n =0 ⇒lim_(n→∞) U_n =∫_0 ^1 (√(1−0))ln^2 xdx=∫_0 ^1 ln^2 x dx x=e^(−t) ,x∈(0,1)⇒t∈(∞,0)⇒dx=−e^(−t) dt ∫_0 ^1 ln^2 x dx=∫_∞ ^( 0) ln^2 (e^(−t) )(−e^t dt)= =∫_0 ^∞ t^2 e^(−t) dt=2 ⇒lim_(n→∞) U_n =2

Un=∫011−xnln2xdx⇒x∈(0,1)⇒limxn→∞n=0Double subscripts: use braces to clarifyx=e−t,x∈(0,1)⇒t∈(∞,0)⇒dx=−e−tdt∫01ln2xdx=∫∞0ln2(e−t)(−etdt)==∫0∞t2e−tdt=2Double subscripts: use braces to clarify

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