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Question Number 143609 by mnjuly1970 last updated on 16/Jun/21

Prove that:: Ω:=∫_0 ^( 1) ((ln^2 (1−x).ln(x))/x)dx=((−1)/2) ζ (4 ) Without using the “Beta function” m.n

Provethat::Ω:=∫01ln2(1−x).ln(x)xdx=−12ζ(4)Withoutusingthe‘‘Betafunction″m.n

Answered by mindispower last updated on 16/Jun/21

∫((ln(1−x))/x)dx=−li_2 (x) =−[li_2 (x)ln(x)ln(1−x)]_0 ^1 −∫_0 ^1 ((li_2 (x))/(1−x))ln(x)dx +∫_0 ^1 ((li_2 (x)ln(1−x))/x)dx =−∫_0 ^1 ((li_2 (x)ln(x))/(1−x))+∫_0 ^1 li_2 (x).(−d(li_2 (x)) =−∫_0 ^1 ((li_2 (x)ln(x))/(1−x))dx−(1/2)ζ^2 (2) −∫_0 ^1 ((li_2 (x)ln(x))/(1−x))dx=−Σ_(n≥1) Σ_(k≥0) ∫_0 ^1 (x^n /n^2 ).x^k ln(x)dx =Σ_(n≥1) Σ_(k≥0) (1/n^2 )∫_0 ^1 x^(n+k) (−ln(x))dx Σ_(n≥1) Σ_(k≥0) (1/(n^2 (n+k+1)))=Σ_(n≥1) Σ_(k≥n+1) (1/(k^2 n^2 ))=S start ζ(2).ζ(2)=Σ_(n≥1) Σ_(k≥1) (1/(n^2 k^2 ))=Σ_(n≥1) Σ_(k≥n+1) (1/(n^2 k^2 ))+Σ_(n≥1) (1/(n^2 .n^2 )) +Σ_(n≥2 ) Σ_(k≤n−1) (1/(n^2 k^2 )) ⇒2Σ_(n≥1) Σ_(k≥n+1) (1/(n^2 k^2 ))+ζ(4)=ζ^2 (2) ⇒S=−((ζ(4))/2)+((ζ^2 (2))/2) Ω=−((ζ(4))/2)+((ζ^2 (2))/2)−((ζ^2 (2))/2)=((ζ(4))/2)

∫ln(1−x)xdx=−li2(x)=−[li2(x)ln(x)ln(1−x)]01−∫01li2(x)1−xln(x)dx+∫01li2(x)ln(1−x)xdx=−∫01li2(x)ln(x)1−x+∫01li2(x).(−d(li2(x))=−∫01li2(x)ln(x)1−xdx−12ζ2(2)−∫01li2(x)ln(x)1−xdx=−∑n⩾1∑k⩾0∫01xnn2.xkln(x)dx=∑n⩾1∑k⩾01n2∫01xn+k(−ln(x))dx∑n⩾1∑k⩾01n2(n+k+1)=∑n⩾1∑k⩾n+11k2n2=Sstartζ(2).ζ(2)=∑n⩾1∑k⩾11n2k2=∑n⩾1∑k⩾n+11n2k2+∑n⩾11n2.n2+∑n⩾2∑k⩽n−11n2k2⇒2∑n⩾1∑k⩾n+11n2k2+ζ(4)=ζ2(2)⇒S=−ζ(4)2+ζ2(2)2Ω=−ζ(4)2+ζ2(2)2−ζ2(2)2=ζ(4)2

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