# Questions tagged [catalans-constant]

For questions about special identities and problems involving Catalan's constant as well as general questions about the constant itself.

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### $I = \int_0^{\pi/2} \left( \int_0^{\pi/2} \frac{\log(\cos(x/2)) - \log(\cos(y/2))}{\cos(x) - \cos(y)} dx\right) \ dy$

\begin{align*} I = \int_0^{\pi/2} \left( \int_0^{\pi/2} \frac{\log(\cos(x/2)) - \log(\cos(y/2))}{\cos(x) - \cos(y)} dx\right) \ dy \end{align*} What I do so far Let $u = \cos(x/2)$ and $v = \cos(y/2)$....
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### Calculating a limit of a sequence of integrals that approaches the catalan constant

Let $f\colon[0, 1] \to \mathbb{R}$, $f(x)=\frac{\arctan(x)}{x}, x \in (0, 1], f(0)=1$. Prove that $$\lim_{n \to \infty} n \left(\frac{\pi}{4}-n\int_0^1\frac{x^n}{1+x^{2n}}dx \right)=\int_0^1f(x)dx$$ ...
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### For $n\ge m\ge 1$, how far can we walk with $\int_0^{\frac{\pi}{2}} \frac{x^n}{\sin^m x} d x$?

In the post, I tackled the integral by power series and integration by parts and obtained that $$\int_0^{\frac{\pi}{2}} \frac{x^2}{\sin x} d x=2\pi G-\frac{7}{2}\zeta(3)$$ where $G$ is the Catalan’s ...
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### Are there any other decent methods to evaluate $\int_0^1 \frac{\ln \left(1-x^4\right)}{1+x^2} d x?$

We first split the integrand into 3 parts as \begin{aligned} \int_0^1 \frac{\ln \left(1-x^4\right)}{1+x^2} d x &= \underbrace{\int_0^1 \frac{\ln \left(1+x^2\right)}{1+x^2} d x}_J+\underbrace{\...
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### Estimate: $\pi \to e \to \log(2) \to G$ by sampling uniform distribution

Successively: $\pi \to e \to \log(2) \to G$ were calculated/estimated by sampling uniform distributions. Method: With a normal distribution $\pi$ can be calculated with help of the PDF (probability ...
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### How to prove $\int_0^1 \frac{\arctan^2(x)\ln\left(\frac{x}{(1-x)^2}\right)}x \, \mathrm{d}x=G^2$?

A while back I made a post asking for examples of integrals which evaluated to famous irrational constants (or constants that were very likely irrational but yet unproven to be). The top answer in ...
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### Can this formula for $\zeta(3)$ be proven or simplified further?

This question is related to the equivalence of formulas (1) and (2) below where formula (1) is from a post on the Harmonic Series Facebook group and formula (2) is based on evaluation of the integral ...
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### How many ways to deal with the integral $\int_{0}^{\frac{\pi}{4}} \ln (\tan x) d x$ and $\int_{0}^{\frac{\pi}{4}} \ln (\sin x) d x$?

After finding that $\displaystyle \int_{0}^{\frac{\pi}{2}} \ln (\tan x) d x =0$ in my post, I was curious about the value of the integral with different upper limit $\dfrac{\pi}{4}$. The answer is ...
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### How do I evaluate $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}H_n}{2n+1}?$

So, I am coming from this question. I managed to bring the answer to this series: $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}H_n}{2n+1}$$ where $H_n$ is the $n$th harmonic number. However, as seen in the ...
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### Finding $\int_0^{\frac{\pi}{2}} \frac{x}{\sin x} dx$

Is there a way to show $$\int_0^{\frac{\pi}{2}} \frac{x}{\sin x} dx = 2C$$ where $C=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^2}$ is Catalan’s constant, preferably without using complex analysis? The ...
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### Question related to potential closed-form representation of Catalan's constant

The motivation for this question is to find a closed-form representation for Catalan's constant. Formula (1) below for the Dirichlet beta function $\beta(s)$ (which I believe is globally convergent) ...
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### Catalan constant's integral representation

Catalan constant is known to have a rich source of integral identities, here is the formula I found: $$\int_0^\infty \frac{\sin^{-1}(\sin(x))}{x} \,dx \ =2G.$$ This can be proved by analyzing the ...
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### Question on Difference Root representations of $\eta(2 n+1)$ and $\beta(2n)$ where $n\in\mathbb{N}$

I've noticed the Dirichlet eta function $\eta(s)$ and the Dirichlet beta function $\beta(s)$ can be represented by difference roots at odd and even positive integers respectfully. For example, ...
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### Computation of $\int_0^1 \frac{\arctan^2 x\ln x}{1+x}dx$

I'm searching for a "simple" proof of: \begin{align}\int_0^1 \frac{\arctan^2 x\ln x}{1+x}dx=-\frac{233}{5760}\pi^4-\frac{5}{48}\pi^2\ln ^2 2+\text{Li}_4\left(\frac{1}{2}\right)+\frac{7}{16}\...
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### Prove $\int_{5\pi/36}^{7\pi/36} \ln (\cot t )dt +\int_{\pi/36}^{3\pi/36} \ln (\cot t )dt = \frac49G$

I have the conjecture for the integral $$\int_{\frac{5\pi}{36}}^{\frac{7\pi}{36}} \ln (\cot t )\>dt +\int_{\frac{\pi}{36}}^{\frac{3\pi}{36}} \ln (\cot t )\>dt = \frac49G$$ where $G$ is the ...
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### Evaluate $\int_0^1 \frac{\ln{\left(ax^2+2bx+a\right)}}{x^2+1} \; \mathrm{d}x, \;a>b>0$
Evaluate $$\int_0^1 \frac{\ln{\left(ax^2+2bx+a\right)}}{x^2+1} \; \mathrm{d}x, \;a>b>0$$ I tried- $$=\frac{\pi \ln{a}}{4}+\int_0^1 \frac{\ln{\left(x^2+2tx+1\right)}}{x^2+1} \; \mathrm{d}x$$ ...