Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 30760 by abdo imad last updated on 25/Feb/18

find I_n = ∫_0 ^1 (lnx)^n dx with n fromN

findIn=∫01(lnx)ndxwithnfromN

Commented by abdo imad last updated on 27/Feb/18

the ch. lnx=−t give I_n = −∫_0 ^(+∞) (−t)^n (−e^(−t) )dt =(−1)^n ∫_0 ^∞ t^n e^(−t) dt =(−1)^n A_(n ) let integrate by parts A_n =[−t^n e^(−t) ]_0 ^∞ −∫_0 ^∞ −nt^(n−1) e^(−t) dt =n∫_0 ^∞ t^(n−1) e^(−t) dt =nA_(n−1) ⇒Π_(k=1) ^n A_k = Π_(k=1) ^n k.Π_(k=1) ^n A_(k−1) ⇒ A_1 .A_2 ....A_n =n! A_0 A_1 ....A_(n−1) ⇒A_n =n! A_0 =n! ⇒ I_n =(−1)^n n! .

thech.lnx=−tgiveIn=−∫0+∞(−t)n(−e−t)dt=(−1)n∫0∞tne−tdt=(−1)nAnletintegratebypartsAn=[−tne−t]0∞−∫0∞−ntn−1e−tdt=n∫0∞tn−1e−tdt=nAn−1⇒∏k=1nAk=∏k=1nk.∏k=1nAk−1⇒A1.A2....An=n!A0A1....An−1⇒An=n!A0=n!⇒In=(−1)nn!.

Terms of Service

Privacy Policy

Contact: info@tinkutara.com