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Prove-that-0-1-ln-2-1-x-ln-x-x-dx-1-2-4-Without-using-the-Beta-function-m-n-




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(1x).ln(x)xdx=12ζ(4)WithoutusingtheBetafunctionm.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(1x)xdx=li2(x)=[li2(x)ln(x)ln(1x)]0101li2(x)1xln(x)dx+01li2(x)ln(1x)xdx=01li2(x)ln(x)1x+01li2(x).(d(li2(x))=01li2(x)ln(x)1xdx12ζ2(2)01li2(x)ln(x)1xdx=n1k001xnn2.xkln(x)dx=n1k01n201xn+k(ln(x))dxn1k01n2(n+k+1)=n1kn+11k2n2=Sstartζ(2).ζ(2)=n1k11n2k2=n1kn+11n2k2+n11n2.n2+n2kn11n2k22n1kn+11n2k2+ζ(4)=ζ2(2)S=ζ(4)2+ζ2(2)2Ω=ζ(4)2+ζ2(2)2ζ2(2)2=ζ(4)2

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