Question

Prove that $\int_0^1 \frac{{\rm{Li}}_2(x)\ln(1-x)\ln^2(x)}{x} \,dx=-\frac{\zeta(6)}{3}$

score 1.1k · Answer 1 · 2026-04-17 10:19:09

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Prove that $\int_0^1 \frac{{\rm{Li}}_2(x)\ln(1-x)\ln^2(x)}{x} \,dx=-\frac{\zeta(6)}{3}$

Published on 17 Apr 2026 - 10:19

score 1.6k · Answer 2 · 2026-04-14 18:31:21

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What is a closed form for ${\large\int}_0^1\frac{\ln^3(1+x)\,\ln^2x}xdx$?

Published on 14 Apr 2026 - 18:31

score 318 · Answer 3 · 2026-04-11 03:33:59

318

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How prime numbers are related to special functions?

Published on 11 Apr 2026 - 3:33

score 435 · Answer 4 · 2026-04-09 01:55:07

435

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Closed-form of a special value dilogarithm identity

Published on 09 Apr 2026 - 1:55

score 467 · Answer 5 · 2026-04-16 13:18:43

467

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Closed-form of $\int_0^1 \operatorname{Li}_p(x) \, dx$

Published on 16 Apr 2026 - 13:18

score 916 · Answer 6 · 2026-04-16 18:43:16

916

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Closed-form of $\int_0^1 \operatorname{Li}_3^3(x)\,dx$ and $\int_0^1 \operatorname{Li}_3^4(x)\,dx$

Published on 16 Apr 2026 - 18:43

score 1.1k · Answer 7 · 2026-04-14 08:08:29

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Closed-forms of real parts of special value dilogarithm identities from inverse tangent integral function

Published on 14 Apr 2026 - 8:08

score 331 · Answer 8 · 2026-04-15 21:46:03

331

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How to prove $\int_{0}^{-1} \frac{\operatorname{Li}_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $

Published on 15 Apr 2026 - 21:46

score 606 · Answer 9 · 2026-04-09 03:31:55

606

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How to integrate the dilogarithms?

Published on 09 Apr 2026 - 3:31

score 384 · Answer 10 · 2026-04-15 23:39:05

384

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Closed- form of $\int_0^1 \frac{{\text{Li}}_3^2(-x)}{x^2}\,dx$

Published on 15 Apr 2026 - 23:39

score 310 · Answer 11 · 2026-05-10 16:49:27

310

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$ \int_0^\infty dx\, x^{-3/4} \ln(1+x) \operatorname{Li}_2 \left( -\frac 1 x \right)$

Published on 10 May 2026 - 16:49

score 516 · Answer 12 · 2026-04-14 21:43:32

516

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Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Published on 14 Apr 2026 - 21:43

score 269 · Answer 13 · 2026-04-10 03:36:14

269

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Hard sum with harmonics numbers

Published on 10 Apr 2026 - 3:36

score 401 · Answer 14 · 2026-04-10 20:57:06

401

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How to evaluate the composed euler sum $\sum_{n=1}^\infty \frac{H_{kn}}{n^2}$

Published on 10 Apr 2026 - 20:57

score 170 · Answer 15 · 2026-04-14 15:19:21

170

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There's a small detail in this proof on why $\sum_{k=1}^{\infty}\frac{1}{k^2} = \frac{\pi^2}{6}$ that I can't figure out

Published on 14 Apr 2026 - 15:19

List Question

Prove that $\int_0^1 \frac{{\rm{Li}}_2(x)\ln(1-x)\ln^2(x)}{x} \,dx=-\frac{\zeta(6)}{3}$

What is a closed form for ${\large\int}_0^1\frac{\ln^3(1+x)\,\ln^2x}xdx$?

How prime numbers are related to special functions?

Closed-form of a special value dilogarithm identity

Closed-form of $\int_0^1 \operatorname{Li}_p(x) \, dx$

Closed-form of $\int_0^1 \operatorname{Li}_3^3(x)\,dx$ and $\int_0^1 \operatorname{Li}_3^4(x)\,dx$

Closed-forms of real parts of special value dilogarithm identities from inverse tangent integral function

How to prove $\int_{0}^{-1} \frac{\operatorname{Li}_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $

How to integrate the dilogarithms?

Closed- form of $\int_0^1 \frac{{\text{Li}}_3^2(-x)}{x^2}\,dx$

$ \int_0^\infty dx\, x^{-3/4} \ln(1+x) \operatorname{Li}_2 \left( -\frac 1 x \right)$

Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Hard sum with harmonics numbers

How to evaluate the composed euler sum $\sum_{n=1}^\infty \frac{H_{kn}}{n^2}$

There's a small detail in this proof on why $\sum_{k=1}^{\infty}\frac{1}{k^2} = \frac{\pi^2}{6}$ that I can't figure out

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