Question

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

score 1k · Answer 1 · 2026-03-25 14:27:16

1k

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

Published on 25 Mar 2026 - 14:27

score 350 · Answer 2 · 2019-07-03 17:58:11

350

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Calculate $\int_0^1\arctan x\left(\frac{3\ln(1+x^2)}{1+x}-\frac{2\ln(1+x)}{x}\right)\ dx$

Published on 03 Jul 2019 - 17:58

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

246

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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 120 · Answer 4 · 2026-03-25 14:24:39

120

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

Published on 25 Mar 2026 - 14:24

score 162 · Answer 5 · 2019-07-06 22:39:31

162

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$A=\frac{1}{1}+\frac{1}{10}+\frac{1}{11}+\dots+\frac{1}{19}+\frac{1}{21}+\frac{1}{31}+\dots+\frac{1}{91}+\frac{1}{100}+\frac{1}{101}+\dots$

Published on 06 Jul 2019 - 22:39

score 190 · Answer 6 · 2019-07-09 15:43:45

190

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Find a closed form of $\sum_{n=1}^{\infty} \frac{{H_{n-1}^{(2)}}x^{2n}}{n^2{{2n}\choose{n}}}$.

Published on 09 Jul 2019 - 15:43

score 188 · Answer 7 · 2026-03-25 16:03:58

188

Views

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

Published on 25 Mar 2026 - 16:03

score 124 · Answer 8 · 2019-07-12 19:53:47

124

Views

Integral form for a Taylor series $\sum _{n=1}^{\infty } \frac{\binom{2 n}{n} \left(4 n H_n-2 n H_{2 n}-1\right) x^{n}}{n!}$

Published on 12 Jul 2019 - 19:53

score 189 · Answer 9 · 2019-07-13 08:56:28

189

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How to prove $ \frac{\ln^k(1+x)}{k!}=\sum_{n=k}^\infty(-1)^{n-k} \begin{bmatrix} n \\ k \end{bmatrix}\frac{x^n}{n!}$

Published on 13 Jul 2019 - 8:56

score 537 · Answer 10 · 2019-07-14 22:52:07

537

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Prove $ \int_0^1 \frac{\ln^a(1-x)\ln(1+x)}{x}dx=(-1)^a a! \sum_{n=1}^\infty\frac{H_n^{(a+1)}}{n2^n}$

Published on 14 Jul 2019 - 22:52

score 963 · Answer 11 · 2026-03-25 19:06:25

963

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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 25 Mar 2026 - 19:06

score 465 · Answer 12 · 2019-07-15 08:40:23

465

Views

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

Published on 15 Jul 2019 - 8:40

score 242 · Answer 13 · 2019-07-16 03:12:54

242

Views

Express $\sum_{j=1}^{n}\sum_{i=1}^{n} \frac{1}{i(i+j)}$ in terms of harmonic numbers

Published on 16 Jul 2019 - 3:12

score 199 · Answer 14 · 2026-03-25 14:25:19

199

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How to find Sum of Sum of Partial Harmonic Series: $\sum_{b=1}^{m} \sum_{k=1}^{b} \frac{1}{k} $?

Published on 25 Mar 2026 - 14:25

score 46 · Answer 15 · 2019-07-18 03:57:46

46

Views

Notation in harmonic series

Published on 18 Jul 2019 - 3:57

List Question

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

Calculate $\int_0^1\arctan x\left(\frac{3\ln(1+x^2)}{1+x}-\frac{2\ln(1+x)}{x}\right)\ dx$

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

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

$A=\frac{1}{1}+\frac{1}{10}+\frac{1}{11}+\dots+\frac{1}{19}+\frac{1}{21}+\frac{1}{31}+\dots+\frac{1}{91}+\frac{1}{100}+\frac{1}{101}+\dots$

Find a closed form of $\sum_{n=1}^{\infty} \frac{{H_{n-1}^{(2)}}x^{2n}}{n^2{{2n}\choose{n}}}$.

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

Integral form for a Taylor series $\sum _{n=1}^{\infty } \frac{\binom{2 n}{n} \left(4 n H_n-2 n H_{2 n}-1\right) x^{n}}{n!}$

How to prove $ \frac{\ln^k(1+x)}{k!}=\sum_{n=k}^\infty(-1)^{n-k} \begin{bmatrix} n \\ k \end{bmatrix}\frac{x^n}{n!}$

Prove $ \int_0^1 \frac{\ln^a(1-x)\ln(1+x)}{x}dx=(-1)^a a! \sum_{n=1}^\infty\frac{H_n^{(a+1)}}{n2^n}$

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

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

Express $\sum_{j=1}^{n}\sum_{i=1}^{n} \frac{1}{i(i+j)}$ in terms of harmonic numbers

How to find Sum of Sum of Partial Harmonic Series: $\sum_{b=1}^{m} \sum_{k=1}^{b} \frac{1}{k} $?

Notation in harmonic series

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