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calculate-0-logx-x-4-x-2-1-dx-




Question Number 136036 by mathmax by abdo last updated on 18/Mar/21
calculate ∫_0 ^∞  ((logx)/(x^4  +x^2  +1))dx
calculate0logxx4+x2+1dx
Commented by Ajetunmobi last updated on 18/Mar/21
  hi sir how can we chat privately ?
hisirhowcanwechatprivately?
Commented by Ajetunmobi last updated on 18/Mar/21
Commented by liki last updated on 19/Mar/21
Drop your whatssap number here !
Dropyourwhatssapnumberhere!
Commented by mathmax by abdo last updated on 31/May/21
you are right liki...
youarerightliki
Answered by Ar Brandon last updated on 18/Mar/21
Δ=∫_0 ^∞ ((lnx)/(x^4 +x^2 +1))dx       =∫_0 ^1 ((lnx)/(x^4 +x^2 +1))dx+∫_1 ^∞ ((lnx)/(x^4 +x^2 +1))dx       =∫_0 ^1 ((lnx)/(x^4 +x^2 +1))dx−∫_0 ^1 ((x^2 lnx)/(x^4 +x^2 +1))dx       =∫_0 ^1 (((1−x^2 )lnx)/((1−x^6 )))dx−∫_0 ^1 ((x^2 (1−x^2 )lnx)/((1−x^6 )))dx       =(1/(36))∫_0 ^1 (((u^(−(5/6)) −u^(−(1/2)) )lnu)/(1−u))du−(1/(36))∫_0 ^1 (((u^(−(1/2)) −u^(−(1/6)) )lnu)/(1−u))du       =−(1/(36)){ψ^1 ((1/6))−ψ^1 ((1/2))−ψ^1 ((1/2))+ψ^1 ((5/6))}       =−(1/(36)){(π^2 /(sin^2 ((π/6))))−2ψ^1 ((1/2))}=−(1/(36)){4π^2 −2×4×(3/4)ζ(2)}       =−(((4π^2 −π^2 ))/(36))=−(π^2 /(12))★
Δ=0lnxx4+x2+1dx=01lnxx4+x2+1dx+1lnxx4+x2+1dx=01lnxx4+x2+1dx01x2lnxx4+x2+1dx=01(1x2)lnx(1x6)dx01x2(1x2)lnx(1x6)dx=13601(u56u12)lnu1udu13601(u12u16)lnu1udu=136{ψ1(16)ψ1(12)ψ1(12)+ψ1(56)}=136{π2sin2(π6)2ψ1(12)}=136{4π22×4×34ζ(2)}=(4π2π2)36=π212
Commented by Dwaipayan Shikari last updated on 18/Mar/21
  😃
😃
Commented by Ajetunmobi last updated on 18/Mar/21
  you did the same solution as mine   but why?
youdidthesamesolutionasminebutwhy?
Commented by Ar Brandon last updated on 18/Mar/21
Because I wanna learn from you too.  Just as I do with the other members.  😄😇
BecauseIwannalearnfromyoutoo.JustasIdowiththeothermembers.😄😇
Commented by Ajetunmobi last updated on 18/Mar/21
  ok sir we are all learnig from each   other
oksirwearealllearnigfromeachother
Commented by Ar Brandon last updated on 18/Mar/21
😀
😀
Commented by SLVR last updated on 13/Apr/21
good evening mr.ar.Brandon   kidly help me what  is ψ′(1/6)..  how it is
goodeveningmr.ar.Brandonkidlyhelpmewhatisψ(1/6)..howitis

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