Question

Prove $\int_{0}^{1} \frac{\sin^{-1}(x)}{x} dx = \frac{\pi}{2}\ln2$

score 466 · Answer 1 · 2016-10-31 02:14:26

466

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Prove $\int_{0}^{1} \frac{\sin^{-1}(x)}{x} dx = \frac{\pi}{2}\ln2$

Published on 31 Oct 2016 - 2:14

score 926 · Answer 2 · 2016-11-02 19:22:51

926

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Closed form for $\int_0^e\mathrm{Li}_2(\ln{x})\,dx$?

Published on 02 Nov 2016 - 19:22

score 106 · Answer 3 · 2016-11-13 16:22:08

106

Views

Proving that swapping the order of this summation is justified

Published on 13 Nov 2016 - 16:22

score 52 · Answer 4 · 2016-11-13 17:43:18

52

Views

Definition of polylog for all complex $\text{z}$

Published on 13 Nov 2016 - 17:43

score 211 · Answer 5 · 2016-12-06 01:47:48

211

Views

Analytically solving an infinite series of periodic functions

Published on 06 Dec 2016 - 1:47

score 1.1k · Answer 6 · 2017-01-07 04:26:25

1.1k

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Evaluate $\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta $

Published on 07 Jan 2017 - 4:26

score 1.7k · Answer 7 · 2017-01-18 14:31:55

1.7k

Views

Equating the integrals of 1/(1-xy) and 2/(1+xy) by elementary calculus?

Published on 18 Jan 2017 - 14:31

score 209 · Answer 8 · 2017-02-10 13:17:35

209

Views

Mathematical reasoning to get closed-forms or nice definite integrals from these outputs of Wolfram Alpha

Published on 10 Feb 2017 - 13:17

score 222 · Answer 9 · 2017-02-14 23:33:44

222

Views

Evaluating Sum involving binomial coefficients and powers

Published on 14 Feb 2017 - 23:33

score 86 · Answer 10 · 2023-02-01 13:14:45

86

Views

Converting a 2D lattice sum into a sum over 1D lattice sums in a circle

Published on 01 Feb 2023 - 13:14

score 359 · Answer 11 · 2023-02-11 22:29:09

359

Views

Imaginary part of dilogarithm

Published on 11 Feb 2023 - 22:29

score 75 · Answer 12 · 2026-02-23 12:02:28

75

Views

Solving a set of implicit equations involving Polylogarithms

Published on 23 Feb 2026 - 12:02

score 92 · Answer 13 · 2023-03-21 15:29:09

92

Views

Asymptotics of an integral involving the exponential integral

Published on 21 Mar 2023 - 15:29

score 86 · Answer 14 · 2023-04-06 17:36:06

86

Views

Finding a recurrence relation to evaluate $\int_{a}^{1}\mathrm{d}x\,\frac{x^{n}}{\sqrt{1-x^{2}}}\ln{\left(\frac{x+a}{x-a}\right)}$

Published on 06 Apr 2023 - 17:36

score 91 · Answer 15 · 2023-04-24 10:21:30

91

Views

Series with power of generalized harmonic number $\displaystyle\sum_{k=1}^{\infty}\left(H_k^{(s)}\right)^n x^k$

Published on 24 Apr 2023 - 10:21

List Question

Prove $\int_{0}^{1} \frac{\sin^{-1}(x)}{x} dx = \frac{\pi}{2}\ln2$

Closed form for $\int_0^e\mathrm{Li}_2(\ln{x})\,dx$?

Proving that swapping the order of this summation is justified

Definition of polylog for all complex $\text{z}$

Analytically solving an infinite series of periodic functions

Evaluate $\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta $

Equating the integrals of 1/(1-xy) and 2/(1+xy) by elementary calculus?

Mathematical reasoning to get closed-forms or nice definite integrals from these outputs of Wolfram Alpha

Evaluating Sum involving binomial coefficients and powers

Converting a 2D lattice sum into a sum over 1D lattice sums in a circle

Imaginary part of dilogarithm

Solving a set of implicit equations involving Polylogarithms

Asymptotics of an integral involving the exponential integral

Finding a recurrence relation to evaluate $\int_{a}^{1}\mathrm{d}x\,\frac{x^{n}}{\sqrt{1-x^{2}}}\ln{\left(\frac{x+a}{x-a}\right)}$

Series with power of generalized harmonic number $\displaystyle\sum_{k=1}^{\infty}\left(H_k^{(s)}\right)^n x^k$

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