Question

Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

score 668 · Answer 1 · 2026-03-27 13:42:16

668

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Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

Published on 27 Mar 2026 - 13:42

score 85 · Answer 2 · 2015-09-02 14:26:57

85

Views

Closed form of integral involving Bessel $J_0$

Published on 02 Sep 2015 - 14:26

score 653 · Answer 3 · 2026-03-26 09:41:05

653

Views

Closed form for $\sum_{k=1}^\infty(\zeta(4k+1)-1)$

Published on 26 Mar 2026 - 9:41

score 356 · Answer 4 · 2015-09-06 04:38:14

356

Views

Summation of series containing logarithm: $\sum_{n=1}^\infty \ln \frac{(n+1)(3n+1)}{n(3n+4)}$

Published on 06 Sep 2015 - 4:38

score 885 · Answer 5 · 2026-03-27 17:06:01

885

Views

Conjecture $\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)=\frac{7\pi^2}{48}-\frac13\arctan^22-\frac16\arctan^23-\frac18\ln^2(\tfrac{18}5)$

Published on 27 Mar 2026 - 17:06

score 90 · Answer 6 · 2015-09-08 14:45:10

90

Views

Closed form expressions for series solutions of D.E's

Published on 08 Sep 2015 - 14:45

score 64 · Answer 7 · 2026-03-31 07:56:49

64

Views

Can anyone identify the function that represents this infinite product?

Published on 31 Mar 2026 - 7:56

score 60 · Answer 8 · 2026-03-26 16:07:02

60

Views

What is the solution of this recursion, that's defined in terms of a sum, but with this $1$ odd twist?

Published on 26 Mar 2026 - 16:07

score 866 · Answer 9 · 2026-03-25 14:25:14

866

Views

Conjecture $\int_0^1\frac{\ln^2\left(1+x+x^2\right)}x dx\stackrel?=\frac{2\pi}{9\sqrt3}\psi^{(1)}(\tfrac13)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3)$

Published on 25 Mar 2026 - 14:25

score 1.2k · Answer 10 · 2026-03-25 14:22:47

1.2k

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Integral ${\large\int}_0^1\ln^3\!\left(1+x+x^2\right)dx$

Published on 25 Mar 2026 - 14:22

score 70 · Answer 11 · 2015-09-10 17:03:05

70

Views

Closed form of a power series

Published on 10 Sep 2015 - 17:03

score 972 · Answer 12 · 2026-03-25 10:56:03

972

Views

Conjectured closed form for $\operatorname{Li}_2\!\left(\sqrt{2-\sqrt3}\cdot e^{i\pi/12}\right)$

Published on 25 Mar 2026 - 10:56

score 426 · Answer 13 · 2015-09-13 06:02:05

426

Views

What's the $n$-th derivative of $\ln(\sin(x))$?

Published on 13 Sep 2015 - 6:02

score 88 · Answer 14 · 2015-09-13 17:23:12

88

Views

Sum of the series $\sum u_n$ where $u_n=\frac{\sqrt{(n-1)!}}{(1+\sqrt{1}) \dots (1+\sqrt{n})}$

Published on 13 Sep 2015 - 17:23

score 970 · Answer 15 · 2015-09-14 10:22:18

970

Views

How to write a closed form expression for $\sum \limits_{i=0}^{n-1}a_i$ in terms of $a_0$ and $\sum \limits_{i=1}^{n}a_i$?

Published on 14 Sep 2015 - 10:22

List Question

Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

Closed form of integral involving Bessel $J_0$

Closed form for $\sum_{k=1}^\infty(\zeta(4k+1)-1)$

Summation of series containing logarithm: $\sum_{n=1}^\infty \ln \frac{(n+1)(3n+1)}{n(3n+4)}$

Conjecture $\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)=\frac{7\pi^2}{48}-\frac13\arctan^22-\frac16\arctan^23-\frac18\ln^2(\tfrac{18}5)$

Closed form expressions for series solutions of D.E's

Can anyone identify the function that represents this infinite product?

What is the solution of this recursion, that's defined in terms of a sum, but with this $1$ odd twist?

Conjecture $\int_0^1\frac{\ln^2\left(1+x+x^2\right)}x dx\stackrel?=\frac{2\pi}{9\sqrt3}\psi^{(1)}(\tfrac13)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3)$

Integral ${\large\int}_0^1\ln^3\!\left(1+x+x^2\right)dx$

Closed form of a power series

Conjectured closed form for $\operatorname{Li}_2\!\left(\sqrt{2-\sqrt3}\cdot e^{i\pi/12}\right)$

What's the $n$-th derivative of $\ln(\sin(x))$?

Sum of the series $\sum u_n$ where $u_n=\frac{\sqrt{(n-1)!}}{(1+\sqrt{1}) \dots (1+\sqrt{n})}$

How to write a closed form expression for $\sum \limits_{i=0}^{n-1}a_i$ in terms of $a_0$ and $\sum \limits_{i=1}^{n}a_i$?

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