Question

Evaluate $\int_0^1\frac{\ln x\ln(1-x)}{1+x^2}\ dx$

score 1k · Answer 1 · 2019-07-02 20:39:26

1k

Views

Evaluate $\int_0^1\frac{\ln x\ln(1-x)}{1+x^2}\ dx$

Published on 02 Jul 2019 - 20:39

score 126 · Answer 2 · 2019-07-03 18:42:14

126

Views

compute the following integral in closed form : $\int_0^{\infty}\frac{\arctan x}{a^2x^2+1}\,dx$

Published on 03 Jul 2019 - 18:42

score 246 · Answer 3 · 2026-03-25 03:03:02

246

Views

On the alternating Euler sum $\sum_{n = 1}^\infty \frac{(-1)^n H_n H_{2n}}{2n + 1}$

Published on 25 Mar 2026 - 3:03

score 391 · Answer 4 · 2019-07-05 19:58:04

391

Views

Powerful Integral $\int_0^x\frac{t\ln(1-t)}{1+t^2}\ dt$

Published on 05 Jul 2019 - 19:58

score 119 · Answer 5 · 2019-07-06 08:18:02

119

Views

Generalized form of this Harmonic Number series $\sum_{n=1}^{\infty} \frac{{H_n}x^{n+1}}{(n+1)^3}$

Published on 06 Jul 2019 - 8:18

score 187 · Answer 6 · 2019-07-12 01:49:54

187

Views

Compute the integral in a closed form : $\int_0^{1}\operatorname{Li}_2(1-x)dx$

Published on 12 Jul 2019 - 1:49

score 186 · Answer 7 · 2019-07-12 06:00:34

186

Views

Unsure about solution to $\int_0^\frac{\pi}{4} \frac{\ln\left| 1 + \tan(x)\right|}{\left( 1 + \tan(x)\right)^n}\:dx$

Published on 12 Jul 2019 - 6:00

score 786 · Answer 8 · 2019-07-12 10:03:06

786

Views

A conjectured value for $\operatorname{Re} \operatorname{Li}_4 (1 + i)$

Published on 12 Jul 2019 - 10:03

score 962 · Answer 9 · 2019-07-15 00:22:05

962

Views

Computing $\int_0^1\frac{\ln^3(1-x)\ln(1+x)}{x}dx$ or $\sum_{n=1}^\infty\frac{H_n^{(4)}}{n2^n}$

Published on 15 Jul 2019 - 0:22

score 179 · Answer 10 · 2019-07-15 12:15:21

179

Views

Generalizing Oksana's trilogarithm relation to $\text{Li}_3(\frac{n}8)$?

Published on 15 Jul 2019 - 12:15

score 169 · Answer 11 · 2019-07-18 11:53:49

169

Views

Expressing $\sum_{n = 1}^\infty \sum_{k = 1}^n \frac{1}{n^4 k\,2^k}$ as a finite sum involving $\zeta(\cdot)$, $Li_k(\cdot)$, $\pi$, and $\ln 2$

Published on 18 Jul 2019 - 11:53

score 226 · Answer 12 · 2019-07-26 15:53:34

226

Views

On the Clausen triple $8\rm{Cl}_2\left(\frac{\pi}2\right)+3\rm{Cl}_2\left(\frac{\pi}3\right)=12\,\rm{Cl}_2\left(\frac{\pi}6\right)$

Published on 26 Jul 2019 - 15:53

score 1.5k · Answer 13 · 2019-07-27 22:17:03

1.5k

Views

Compute $\int_0^{1/2}\frac{\left(\operatorname{Li}_2(x)\right)^2}{x}dx$ or $\sum_{n=1}^\infty \frac{H_n^{(2)}}{n^32^n}$

Published on 27 Jul 2019 - 22:17

score 381 · Answer 14 · 2019-07-30 08:46:12

381

Views

Challenging sum: Calculate $\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^52^k{2k \choose k}}$

Published on 30 Jul 2019 - 8:46

score 167 · Answer 15 · 2026-03-25 06:50:34

167

Views

What is the series $\sum_{n=1}^{\infty} \frac{e^{-n^2 x}}{n}$?

Published on 25 Mar 2026 - 6:50

List Question

Evaluate $\int_0^1\frac{\ln x\ln(1-x)}{1+x^2}\ dx$

compute the following integral in closed form : $\int_0^{\infty}\frac{\arctan x}{a^2x^2+1}\,dx$

On the alternating Euler sum $\sum_{n = 1}^\infty \frac{(-1)^n H_n H_{2n}}{2n + 1}$

Powerful Integral $\int_0^x\frac{t\ln(1-t)}{1+t^2}\ dt$

Generalized form of this Harmonic Number series $\sum_{n=1}^{\infty} \frac{{H_n}x^{n+1}}{(n+1)^3}$

Compute the integral in a closed form : $\int_0^{1}\operatorname{Li}_2(1-x)dx$

Unsure about solution to $\int_0^\frac{\pi}{4} \frac{\ln\left| 1 + \tan(x)\right|}{\left( 1 + \tan(x)\right)^n}\:dx$

A conjectured value for $\operatorname{Re} \operatorname{Li}_4 (1 + i)$

Computing $\int_0^1\frac{\ln^3(1-x)\ln(1+x)}{x}dx$ or $\sum_{n=1}^\infty\frac{H_n^{(4)}}{n2^n}$

Generalizing Oksana's trilogarithm relation to $\text{Li}_3(\frac{n}8)$?

Expressing $\sum_{n = 1}^\infty \sum_{k = 1}^n \frac{1}{n^4 k\,2^k}$ as a finite sum involving $\zeta(\cdot)$, $Li_k(\cdot)$, $\pi$, and $\ln 2$

On the Clausen triple $8\rm{Cl}_2\left(\frac{\pi}2\right)+3\rm{Cl}_2\left(\frac{\pi}3\right)=12\,\rm{Cl}_2\left(\frac{\pi}6\right)$

Compute $\int_0^{1/2}\frac{\left(\operatorname{Li}_2(x)\right)^2}{x}dx$ or $\sum_{n=1}^\infty \frac{H_n^{(2)}}{n^32^n}$

Challenging sum: Calculate $\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^52^k{2k \choose k}}$

What is the series $\sum_{n=1}^{\infty} \frac{e^{-n^2 x}}{n}$?

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