Question

Finding $\int_{1}^{\infty} \frac{1}{1+x^2} \frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{\pi^2+\ln^2\left(\frac{x-1}{2}\right)}\text{d}x$

score 480 · Answer 1 · 2022-01-21 02:14:43

480

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Finding $\int_{1}^{\infty} \frac{1}{1+x^2} \frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{\pi^2+\ln^2\left(\frac{x-1}{2}\right)}\text{d}x$

Published on 21 Jan 2022 - 2:14

score 214 · Answer 2 · 2022-02-10 21:45:58

214

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Can this formula for $\zeta(3)$ be proven or simplified further?

Published on 10 Feb 2022 - 21:45

score 1.3k · Answer 3 · 2022-03-16 23:40:21

1.3k

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How to prove $\int_0^1 \frac{\arctan^2(x)\ln\left(\frac{x}{(1-x)^2}\right)}x \, \mathrm{d}x=G^2$?

Published on 16 Mar 2022 - 23:40

score 199 · Answer 4 · 2026-02-23 21:14:34

199

Views

Estimate: $\pi \to e \to \log(2) \to G$ by sampling uniform distribution

Published on 23 Feb 2026 - 21:14

score 163 · Answer 5 · 2022-03-18 14:47:56

163

Views

How to find the closed form for the integral $I_n:=\int_{0}^{1} \frac{x^{n} \ln x}{1+x^{2}} d x, {} $ where $ n\in N$?

Published on 18 Mar 2022 - 14:47

score 123 · Answer 6 · 2022-03-23 07:10:06

123

Views

Narrower the limits harder the integral!

Published on 23 Mar 2022 - 7:10

score 199 · Answer 7 · 2022-04-16 02:35:01

199

Views

Is there any other method to compute $\int_{0}^{\frac{\pi}{4}} y \ln (\cos y) d y$?

Published on 16 Apr 2022 - 2:35

score 126 · Answer 8 · 2022-04-16 03:03:43

126

Views

How to tackle $\int_{0}^{\frac{\pi}{2}}y \ln (1+\cos y)\,d y$?

Published on 16 Apr 2022 - 3:03

score 215 · Answer 9 · 2022-05-31 21:35:15

215

Views

The integral $\int_{0}^{1} \frac{ \log (1-x)}{1+x^2}dx$

Published on 31 May 2022 - 21:35

score 107 · Answer 10 · 2022-07-11 23:26:50

107

Views

Showing $\lim_{x\rightarrow 1^-} \sum_{k=0}^\infty (-x)^k (2k+1) \ln(2k+1) = \frac{-2C}{\pi}$

Published on 11 Jul 2022 - 23:26

score 137 · Answer 11 · 2022-07-28 08:27:36

137

Views

Evaluating $\int_{0}^{\infty} \frac{x}{\cosh (a x)} d x$ for a positive constant $a$

Published on 28 Jul 2022 - 8:27

score 280 · Answer 12 · 2022-07-28 16:13:20

280

Views

Hard integral $\int_{0}^{\frac{\pi}{4}} \ln ^{n}(\tan \theta) d \theta$?

Published on 28 Jul 2022 - 16:13

score 495 · Answer 13 · 2022-08-16 03:29:24

495

Views

Is it hard to evaluate $\int_{0}^{\infty} \frac{\ln \left(1-x+x^{2}\right)}{1+x^{2}} d x$?

Published on 16 Aug 2022 - 3:29

score 280 · Answer 14 · 2022-08-16 09:02:57

280

Views

Computing $\int_{0}^{\pi} \ln (\sin x+2) d x$ and $\int_{0}^{\pi} \ln (2-\sin x) d x$

Published on 16 Aug 2022 - 9:02

score 66 · Answer 15 · 2022-09-14 21:08:38

66

Views

Contour integral for $f(z)=\frac{1}{2}\frac{e^{2z}}{\cosh(e^z)}$

Published on 14 Sep 2022 - 21:08

List Question

Finding $\int_{1}^{\infty} \frac{1}{1+x^2} \frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{\pi^2+\ln^2\left(\frac{x-1}{2}\right)}\text{d}x$

Can this formula for $\zeta(3)$ be proven or simplified further?

How to prove $\int_0^1 \frac{\arctan^2(x)\ln\left(\frac{x}{(1-x)^2}\right)}x \, \mathrm{d}x=G^2$?

Estimate: $\pi \to e \to \log(2) \to G$ by sampling uniform distribution

How to find the closed form for the integral $I_n:=\int_{0}^{1} \frac{x^{n} \ln x}{1+x^{2}} d x, {} $ where $ n\in N$?

Narrower the limits harder the integral!

Is there any other method to compute $\int_{0}^{\frac{\pi}{4}} y \ln (\cos y) d y$?

How to tackle $\int_{0}^{\frac{\pi}{2}}y \ln (1+\cos y)\,d y$?

The integral $\int_{0}^{1} \frac{ \log (1-x)}{1+x^2}dx$

Showing $\lim_{x\rightarrow 1^-} \sum_{k=0}^\infty (-x)^k (2k+1) \ln(2k+1) = \frac{-2C}{\pi}$

Evaluating $\int_{0}^{\infty} \frac{x}{\cosh (a x)} d x$ for a positive constant $a$

Hard integral $\int_{0}^{\frac{\pi}{4}} \ln ^{n}(\tan \theta) d \theta$?

Is it hard to evaluate $\int_{0}^{\infty} \frac{\ln \left(1-x+x^{2}\right)}{1+x^{2}} d x$?

Computing $\int_{0}^{\pi} \ln (\sin x+2) d x$ and $\int_{0}^{\pi} \ln (2-\sin x) d x$

Contour integral for $f(z)=\frac{1}{2}\frac{e^{2z}}{\cosh(e^z)}$

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