Linked Questions

33
votes
7answers
3k views

Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$

I am trying to prove that \begin{equation} \int_{0}^{1}\frac{\log\left(x\right) \log\left(\,{1 - x^{4}}\,\right)}{1 + x^{2}} \,\mathrm{d}x = \frac{\pi^{3}}{16} - 3\mathrm{G}\log\left(2\right) \tag{1} \...
39
votes
4answers
2k views

Proving that $\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+x^2}{(1-x)^2}\right)dx=\frac{\pi^3}{16}$

The following integral was proposed by Cornel Ioan Valean and appeared as Problem $12054$ in the American Mathematical Monthly earlier this year. Prove $$\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+...
25
votes
2answers
3k views

Evaluating $\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$

I am trying to prove that $$\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx = -\frac{\pi^3}{48}-\frac{\pi}{8}\log^2 2 +G\log 2$$ where $G$ is the Catalan's Constant. Numerically, it's ...
32
votes
2answers
2k views

Remarkable logarithmic integral $\int_0^1 \frac{\log^2 (1-x) \log^2 x \log^3(1+x)}{x}dx$

We have the following result ($\text{Li}_{n}$ being the polylogarithm): $$\tag{*}\small{ \int_0^1 \log^2 (1-x) \log^2 x \log^3(1+x) \frac{dx}{x} = -168 \text{Li}_5(\frac{1}{2}) \zeta (3)+96 \text{Li}...
12
votes
2answers
417 views

Integral $\int_0^1 \ln\left(\frac{1-x}{1+x}\right)\ln\left(\frac{1-x^2}{1+x^2}\right)\frac{dx}{x}$

Greetings I saw here (among the last integrals) that: $$\int_0^1 \ln\left(\frac{1-x}{1+x}\right)\ln\left(\frac{1-x^2}{1+x^2}\right)\frac{dx}{x}=\pi C$$ Where $C$ is Catalan's constant. Did this ...
6
votes
3answers
356 views

Find : $\int_0^{\pi/4}x\ln(\sin x)\mathrm dx$

I'm try to find this integral $$\int_0^{\pi/4}x\ln(\sin x)\mathrm dx$$ My try use : $\ln(\sin x)=-\ln2-\sum\limits_{n=1}^{\infty}\frac{\cos (2nx)}{n}$ But I don't know how to complete summation ......
3
votes
4answers
1k views

Evaluating $\int_0^{\pi/4} \ln(\tan x)\ln(\cos x-\sin x)dx=\frac{G\ln 2}{2}$

In order to compute, in an elementary way, $\displaystyle \int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$ (see Evaluating $\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^...
5
votes
5answers
523 views

Integral $\int_0^1 \frac{\ln(1-x)\ln(1+x^2)}{x}dx$

I am trying to solve by a different approach the fourth sum from here, namely: $$S= \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(4n+m)} =\int_0^1 \frac{\ln(1-x)\ln(1-x^4)}{x}dx= \frac{67}{32} \...
3
votes
3answers
489 views

Compute $ \int_0^1\frac{\ln^2(1+x)}{1+x^2}\, dx$

How to prove $$I=\int_0^1\frac{\ln^2(1+x)}{1+x^2}\ dx=4\Im\operatorname{Li}_3(1+i)-\frac{7\pi^3}{64}-\frac{3\pi}{16}\ln^22-2\ln2\ G$$ Where $ \operatorname{Li}_3(x)$ is the the trilogarithm ...
4
votes
3answers
293 views

prove $\ln(1+x^2)\arctan x=-2\sum_{n=1}^\infty \frac{(-1)^n H_{2n}}{2n+1}x^{2n+1}$

I was able to prove the above identity using 1) Cauchy Product of Power series and 2) integration but the point of posting it here is to use it as a reference in our solutions. other approaches ...
6
votes
4answers
374 views

Evaluating $\int_0^1\arctan x\ln(1+x)\left(\frac2x-\frac3{1+x}\right)dx$

How can we find the value of $$\int_0^1\arctan x\ln(1+x)\left(\frac2x-\frac3{1+x}\right)dx$$ using elementary methods? With some help of calculator I get the result: $\displaystyle{\frac3{128}\pi^3-\...
14
votes
0answers
1k views

An interesting identity involving powers of $\pi$ and values of $\eta(s)$ [duplicate]

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
6
votes
1answer
208 views

Integral $\int_0^1\frac{\ln(1+x)}{1+x^2}\left(\frac{3\arctan x}{x}+2\ln x\right)dx$

I was playing around with PARI GP to generate a challenging integral. The method: The function lindep is intended to detect integer dependence. The constants used: $\text{G}\ln 2,\pi^3,\pi\ln^2 2,\...