Prove-that-0-1-ln-2-1-x-ln-x-x-dx-1-2-4-Without-using-the-Beta-function-m-n- Tinku Tara June 3, 2023 Differentiation 0 Comments FacebookTweetPin Question Number 143609 by mnjuly1970 last updated on 16/Jun/21 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)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 Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: expressing-P-x-x-2-x-x-3-x-2-2-in-partial-fractions-gives-A-A-x-3-Bx-C-x-2-2-B-A-x-3-B-x-2-C-x-2-C-A-x-3-B-x-2-C-x-2-DNext Next post: prove-that-lim-x-2-x-2- Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.