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 481 · Answer 1 · 2026-03-25 10:25:08

481

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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 25 Mar 2026 - 10:25

score 215 · Answer 2 · 2026-03-25 11:54:09

215

Views

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

Published on 25 Mar 2026 - 11:54

score 1.3k · Answer 3 · 2026-03-25 10:25:28

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 25 Mar 2026 - 10:25

score 200 · Answer 4 · 2026-04-11 22:11:01

200

Views

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

Published on 11 Apr 2026 - 22:11

score 164 · Answer 5 · 2026-03-25 10:24:13

164

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 25 Mar 2026 - 10:24

score 124 · Answer 6 · 2026-03-25 11:53:23

124

Views

Narrower the limits harder the integral!

Published on 25 Mar 2026 - 11:53

score 200 · Answer 7 · 2026-03-25 10:36:23

200

Views

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

Published on 25 Mar 2026 - 10:36

score 127 · Answer 8 · 2026-03-25 10:36:18

127

Views

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

Published on 25 Mar 2026 - 10:36

score 216 · Answer 9 · 2026-03-25 11:54:09

216

Views

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

Published on 25 Mar 2026 - 11:54

score 108 · Answer 10 · 2026-03-25 10:36:03

108

Views

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

Published on 25 Mar 2026 - 10:36

score 138 · Answer 11 · 2026-03-25 11:54:14

138

Views

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

Published on 25 Mar 2026 - 11:54

score 281 · Answer 12 · 2026-03-25 12:03:14

281

Views

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

Published on 25 Mar 2026 - 12:03

score 496 · Answer 13 · 2026-03-25 10:35:05

496

Views

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

Published on 25 Mar 2026 - 10:35

score 281 · Answer 14 · 2026-03-25 08:54:10

281

Views

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

Published on 25 Mar 2026 - 8:54

score 67 · Answer 15 · 2026-03-25 10:35:36

67

Views

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

Published on 25 Mar 2026 - 10:35

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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