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find-0-ln-1-t-2-t-2-dt-




Question Number 32478 by prof Abdo imad last updated on 25/Mar/18
find  ∫_0 ^∞  ln(((1+t^2 )/t^2 ))dt
find0ln(1+t2t2)dt
Commented by prof Abdo imad last updated on 30/Mar/18
I = ∫_0 ^(+∞)  ln(1+(1/t^2 ))dt  by parts  I = ][t ln(1+(1/t^2 ))]_0 ^(+∞)   − ∫_0 ^(+∞)  t  ((−2)/t^3 ) (1+(1/t^2 ))^(−1) dt  =  2 ∫_0 ^∞    (1/t^2 )  . (1/(1+(1/t^2 )))dt = 2 ∫_0 ^∞   (dt/(t^2  +1)) = 2(π/2)  =π  let prove that lim_(t→+∞)  tln(1+(1/t^2 ))=0  = lim_(u →0)  (1/u) ln (1+u^2 ) =lim_(u→∞)  u ((ln(1+u^2 ))/u^2 ) =0  also lim_(t→o) t ln(1 +(1/t^2 )) =lim_(t→0)  tln(1+t^2 ) −2tlnt  =0  finally   I  =π
I=0+ln(1+1t2)dtbypartsI=][tln(1+1t2)]0+0+t2t3(1+1t2)1dt=201t2.11+1t2dt=20dtt2+1=2π2=πletprovethatlimt+tln(1+1t2)=0=limu01uln(1+u2)=limuuln(1+u2)u2=0alsolimtotln(1+1t2)=limt0tln(1+t2)2tlnt=0finallyI=π

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