Mathematics

Questions tagged [polylogarithmic-integral]

46 questions
46
votes
7 answers

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 $$\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+x^2}{(1-x)^2}\right)dx=\frac{\pi^3}{16}$$ I had…
Zacky
  • 30,834
34
votes
9 answers

Evaluating $\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$

In this thread a friend posted the following integral $$I=\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$$ The best we could do is expressing it in terms of Euler sums $$I=-\frac{\zeta^2(2)}{2}+ \sum_{n\geq 1}\frac{(-1)^{n-1}}{n^2}…
Zaid Alyafeai
  • 14,883
33
votes
7 answers

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} \end{equation} where $\mathrm{G}$ is Catalan's…
Shobhit Bhatnagar
  • 11,906
30
votes
4 answers

How to evaluate $\int_{0}^{1}{\frac{x\ln(1+x)\ln(1+x^2)}{1+x^2}}dx$

How to evaluate $$\int_{0}^{1}{\frac{x\ln(1+x)\ln(1+x^2)}{1+x^2}}dx$$ My attempt $$\begin{align}\int_{0}^{1}{\frac{x\ln(1+x)\ln(1+x^2)}{1+x^2}}dx&=\frac{1}{4}\int_{0}^{1}{\ln(1+x)}d(\ln^2(1+x^2))\\…
Li Taiji
  • 873
29
votes
3 answers

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 value is $-0.199739$. Using the substitution…
Shobhit Bhatnagar
  • 11,906
29
votes
3 answers

Closer form for $\int_0^\infty\frac{(\arctan{x})^2\log^2({1+x^2})}{x^2}dx$

I Would like to know the value of this integral. $$\int_0^\infty\frac{(\arctan{x})^2\log^2({1+x^2})}{x^2}dx$$ I think…
user178256
  • 5,861
25
votes
11 answers

A challenging logarithmic integral $\int_0^1 \frac{\log(1+x)\log(1-x)}{1+x}dx$

While playing around with Mathematica , I found that $$\int_0^1 \frac{\log(1+x)\log(1-x)}{1+x}dx = \frac{1}{3}\log^3(2)-\frac{\pi^2}{12}\log(2)+\frac{\zeta(3)}{8}$$ Please help me prove this result.
Shobhit Bhatnagar
  • 11,906
24
votes
5 answers

Integral ${\large\int}_0^1\ln(1-x)\ln(1+x)\ln^2x\,dx$

This problem was posted at I&S a week ago, and no attempts to solve it have been posted there yet. It looks very alluring, so I decided to repost it here: Prove: \begin{align} &\int_0^1\ln(1-x)\ln(1+x)\ln^2x\,dx \\[2mm] = & \…
Oksana Gimmel
  • 1
21
votes
7 answers

Evaluating $\int_0^1 \frac{\arctan x \log x}{1+x}dx$

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 $\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$ ) i need to show, in a simple way , that:…
FDP
  • 16,180
20
votes
3 answers

Solving $\int_0^1\log^2(1-x)\log^2(1+x)dx$

I came across this problem in the book (Almost) Impossible Integrals, Sums and Series Problem 1.8 $$\int_0^1\log^2(1-x)\log^2(1+x)dx=24-8\zeta(2)-8\zeta(3)-\zeta(4)+8\log2\zeta(2)-4\log^22\zeta(2)+8\log2\zeta(3)-24\log2+12\log^22-4\log^32+\log^42$$…
Souparna
  • 1,082
20
votes
7 answers

Difficult integral $\int_0^{\infty}\frac {\arctan x\log(1+x^2)}{x(1+x^2)}dx=\frac {\pi}2\log^22$

Question: Show that $$\int\limits_0^{\infty}\frac {\arctan x\log(1+x^2)}{x(1+x^2)}dx=\frac {\pi}2\log^22.$$ I can't tell if I'm being an idiot, or if this is a lot more difficult than it looks. First, I tried integration by parts using the fact that…
Frank
  • 6,614
18
votes
8 answers

About the integral $\int_{0}^{1}\frac{\log(x)\log^2(1+x)}{x}\,dx$

I came across the following Integral and have been completely stumped by it. $$\large\int_{0}^{1}\dfrac{\log(x)\log^2(1+x)}{x}dx$$ I'm extremely sorry, but the only thing I noticed was that the limits of the Integral were similar to the Beta…
Make a Difference
  • 339
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15
votes
7 answers

Integral $ \int_0^1 \frac{\ln x \ln(1+x)}{1+x}dx=-\frac{\zeta(3)}{8} $

How can we prove $$ \int_0^1 \frac{\ln x \ln(1+x)}{1+x}dx=-\frac{\zeta(3)}{8}? $$ This has been one of the integrals that came out of an integral from another post on here, but no solution to it. I am not sure how to use a taylor series expansion…
Jeff Faraci
  • 10,575
14
votes
1 answer

How to compute $\int_0^1\frac{\ln x}{x^2-x-1}dx$ without using polylogarithmic function $\text{Li}_2(z)$

How to compute this integral $$\int_0^1\frac{\ln x}{x^2-x-1}dx$$ One method I have thought before is to solve the equation $x^2-x-1=0$ for its roots $x_{1,2}=\frac{\sqrt{5}\pm 1}{2}$ , with $\phi=\frac{\sqrt{5}+ 1}{2}$ . Then, use the…
mbfkk
  • 1,449
13
votes
1 answer

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

I tried to solve this integral and got it, I showed firstly $$\int_0^1 \frac{\ln^2(x+1)+\ln^2\left(\frac{x}{x+1}\right)}{x^2+1} dx=2\Im\left[\text{Li}_3(1+i) \right] $$ and for other integral $$\int_0^1 \frac{\ln\left(\frac{2x}{x^2+1}\right)\ln…
Faoler
  • 2,894
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