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 40887 by prof Abdo imad last updated on 28/Jul/18

calculate ∫_0 ^1 ((tln(t))/(t^2 −1))dt

calculate∫01tln(t)t2−1dt

Commented by prof Abdo imad last updated on 30/Jul/18

let I = ∫_0 ^1 ((tln(t))/(t^2 −1))dt I =−∫_0 ^1 tln(t)(Σ_(n=0) ^∞ t^(2n) )dt =−Σ_(n=0) ^∞ ∫_0 ^1 t^(2n+1) ln(t)dt by parts A_n =∫_0 ^1 t^(2n+1) ln(t)dt =[(1/(2n+2))t^(2n+2) ln(t)]_0 ^1 −∫_0 ^1 (1/(2n+2)) t^(2n+1) dt =−(1/((2n+2)^2 )) ⇒ I =Σ_(n=0) ^∞ (1/((2n+2)^2 )) =(1/4) Σ_(n=0) ^∞ (1/((n+1)^2 )) =(1/4)Σ_(n=1) ^∞ (1/n^2 ) =(1/4) .(π^2 /6) ⇒ I =(π^2 /(24))

letI=∫01tln(t)t2−1dtI=−∫01tln(t)(∑n=0∞t2n)dt=−∑n=0∞∫01t2n+1ln(t)dtbypartsAn=∫01t2n+1ln(t)dt=[12n+2t2n+2ln(t)]01−∫0112n+2t2n+1dt=−1(2n+2)2⇒I=∑n=0∞1(2n+2)2=14∑n=0∞1(n+1)2=14∑n=1∞1n2=14.π26⇒I=π224

Terms of Service

Privacy Policy

Contact: info@tinkutara.com