Question

A closed form for: $\int_{0}^{\infty} \frac{1}{(x-\log x)^2}dx$

score 1.1k · Answer 1 · 2019-06-29 09:44:57

1.1k

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A closed form for: $\int_{0}^{\infty} \frac{1}{(x-\log x)^2}dx$

Published on 29 Jun 2019 - 9:44

score 183 · Answer 2 · 2026-03-25 19:32:48

183

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Prove that : $\int_0^{1}\frac{1-\sqrt{1-x^{4}}}{x^{2}\sqrt{1-x^4}}dx=1-\frac{\sqrt{2}\pi^{\frac{3}{2}}}{\Gamma(\frac{1}{4})^2}$

Published on 25 Mar 2026 - 19:32

score 244 · Answer 3 · 2019-06-30 05:46:48

244

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On closed forms for the binomial sum $\sum_{n=1}^\infty \frac{z^n}{n^p\,\binom {2n}n}$ for general $p$?

Published on 30 Jun 2019 - 5:46

score 510 · Answer 4 · 2026-03-25 19:05:11

510

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

Published on 25 Mar 2026 - 19:05

score 78 · Answer 5 · 2019-06-30 18:22:03

78

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compute the following integral in closed form : $\int_0^{\infty}\frac{\ln(1+x^b)}{x^2+ax+1})dx$

Published on 30 Jun 2019 - 18:22

score 735 · Answer 6 · 2026-03-25 19:06:02

735

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Compute $ \int_0^1\frac{\ln^2(1+x)}{1+x^2}\, dx$

Published on 25 Mar 2026 - 19:06

score 134 · Answer 7 · 2019-07-01 08:15:48

134

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Solving recurrence relations with negative powers or reciprocals

Published on 01 Jul 2019 - 8:15

score 155 · Answer 8 · 2019-07-01 10:50:11

155

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Closed form for the integral $\int_{-1}^{1} \frac{\ln[(1+x^2)~ -x\sqrt{k+x^2}]}{\sqrt{1-x^2}}\mathrm dx$

Published on 01 Jul 2019 - 10:50

score 390 · Answer 9 · 2019-07-01 11:35:37

390

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Integral $\int_0^{1}\frac{(1+x^2)\ln(1+x^4)}{1+x^4}dx$

Published on 01 Jul 2019 - 11:35

score 169 · Answer 10 · 2026-03-25 15:44:07

169

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Any suggestions on a closed form of $\sum_{a=1}^{N} \left({\lfloor{\frac{N+1}{a}}\rfloor+\lfloor{\frac{N-1}{a}}\rfloor}\right)$

Published on 25 Mar 2026 - 15:44

score 126 · Answer 11 · 2019-07-01 19:34:14

126

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Closed form solution for a simple-looking Euclidean geometry problem?

Published on 01 Jul 2019 - 19:34

score 286 · Answer 12 · 2019-07-03 09:59:46

286

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compute the following integral in closed form : $\int_0^{\frac{π}{2}}\frac{x}{(1+\sqrt 2)\sin^{2}(x)+8\cos^{2} x}dx$

Published on 03 Jul 2019 - 9:59

score 351 · Answer 13 · 2026-03-26 02:57:52

351

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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 26 Mar 2026 - 2:57

score 127 · Answer 14 · 2026-03-25 19:10:11

127

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

Published on 25 Mar 2026 - 19:10

score 246 · Answer 15 · 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

List Question

A closed form for: $\int_{0}^{\infty} \frac{1}{(x-\log x)^2}dx$

Prove that : $\int_0^{1}\frac{1-\sqrt{1-x^{4}}}{x^{2}\sqrt{1-x^4}}dx=1-\frac{\sqrt{2}\pi^{\frac{3}{2}}}{\Gamma(\frac{1}{4})^2}$

On closed forms for the binomial sum $\sum_{n=1}^\infty \frac{z^n}{n^p\,\binom {2n}n}$ for general $p$?

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

compute the following integral in closed form : $\int_0^{\infty}\frac{\ln(1+x^b)}{x^2+ax+1})dx$

Compute $ \int_0^1\frac{\ln^2(1+x)}{1+x^2}\, dx$

Solving recurrence relations with negative powers or reciprocals

Closed form for the integral $\int_{-1}^{1} \frac{\ln[(1+x^2)~ -x\sqrt{k+x^2}]}{\sqrt{1-x^2}}\mathrm dx$

Integral $\int_0^{1}\frac{(1+x^2)\ln(1+x^4)}{1+x^4}dx$

Any suggestions on a closed form of $\sum_{a=1}^{N} \left({\lfloor{\frac{N+1}{a}}\rfloor+\lfloor{\frac{N-1}{a}}\rfloor}\right)$

Closed form solution for a simple-looking Euclidean geometry problem?

compute the following integral in closed form : $\int_0^{\frac{π}{2}}\frac{x}{(1+\sqrt 2)\sin^{2}(x)+8\cos^{2} x}dx$

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

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

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

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