Questions tagged [polylogarithmic-integral]
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13 questions
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Integrating $\int_0^1 \frac{\ln x \ln (1+x) \ln(1-x)}{x^{\frac12}} \,dx$
$$\int_0^1 \frac{\ln x \ln (1+x) \ln(1-x)}{x^{\frac12}}
\,dx$$
Now I have seen a similar integral here,
I am unable to deal with the $x^{\frac12}$ term in the denominator.
Here's one idea,
$$\int_0^1x^...
- 5,648
3
votes
3
answers
272
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Evaluating $\int_0^1 \frac{\tan^{-1}(x)\ln^2(x)}{1+x}\,dx$
$$I=\int_0^1 \frac{\tan^{-1}(x)\ln^2(x)}{1+x}\,dx$$
I have not been able to find any similar integrals to refer and solve this one.
$$\int_0^1 \frac{\tan^{-1}(x)\ln^2(x)}{1+x}\,dx\underset{ibp}=-\...
- 5,648
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Tricky integrals: $\int_1^x \frac{\text{Li}_2(t) \log (t+1)}{t}$ and $\int_1^x\frac{\text{Li}_2(t) \log (t)}{t+1}$
I have tried really hard to compute the following integrals, but got nowhere:
Does anyone know a method that could work here ?
The integrals are
$$
\int_{1}^{x}\frac{\operatorname{Li}_{2}\left(t\...
- 39
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votes
5
answers
487
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How to evaluate $\int_0^1 \frac{\arctan(x)}{x} \ln^2(1 - x) \, dx$
Question
How to evaluate $$\int_0^1 \frac{\arctan(x)}{x} \ln^2(1 - x) \, dx$$
My attempt
\begin{align}
\int_0^1 \frac{\arctan(x)}{x} \ln^2(1 - x) \, dx &= \int_0^1 \int_0^1 \frac{\ln^2(1 - x)}{...
- 5,957
20
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3
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779
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Solving $\int_0^1\log^2(1-x)\log^2(1+x)dx$
I came across this problem in the book (Almost) Impossible Integrals, Sums and Series Problem 1.8
$$\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\...
- 1,062
4
votes
4
answers
335
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Nice pair of trilog integrals $\int_0^z \frac{\log ^2(x) \log (1\pm x)}{1\mp x} \, dx$
Recently, in the wake of the solution of Compute $\int_0^1\frac{\ln(1-x)\ln(1+x)}{1+x}\ln\left(\frac{1+x}{2}\right)\ dx$, I stumbled on this symmetric pair of integrals
$$i_{\pm}(z) = \int_0^z \frac{\...
8
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2
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Evaluate $\int_0^1 \frac{\arctan x\ln^2 x}{1+x^2}\,dx$
Empirically, i have obtained the following value:
\begin{align}K&=\int_0^1 \frac{\arctan x\ln^2 x}{1+x^2}\,dx\\
&=\frac{151}{11520}\pi^4-\frac{1}{24}\ln^4 2-\text{Li}_4\left(\frac{1}{2}\right)+...
- 15.1k
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8
answers
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How to evaluate $\int_0^1\frac{\ln x\ln^2(1-x)}{1+x}dx$ in an elegant way?
How to prove, in an elegant way that
$$I=\int_0^1\frac{\ln x\ln^2(1-x)}{1+x}dx=\frac{11}{4}\zeta(4)-\frac14\ln^42-6\operatorname{Li}_4\left(\frac12\right)\ ?$$
First, let me show you how I did it
...
- 26.6k
18
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8
answers
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About the integral $\int_{0}^{1}\frac{\log(x)\log^2(1+x)}{x}\,dx$
I came across the following Integral and have been completely stumped by it.
$$\large\int_{0}^{1}\dfrac{\log(x)\log^2(1+x)}{x}dx$$
I'm extremely sorry, but the only thing I noticed was that the ...
4
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1
answer
304
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Logarithmic twin integrals $\int_0^1\frac{\ln x\ln(1+x^2)}{1\pm x}dx$
Here is what I have done
\begin{align}
&\int_0^1\frac{\ln(x)\ln(1+x^2)}{1-x}dx\\
=&\int_0^1\frac{(1+x)(1+x^2) \log(x)\log(1-x^4)}{1-x^4} \ dx\\
&-\int_0^1\frac{(1+x)\log(x)\log(1-x^2)}{1-x^...
- 44.6k
33
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5
answers
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Integral ${\large\int}_0^1\ln(1-x)\ln(1+x)\ln^2x\,dx$
This problem was posted at I&S a week ago, and no attempts to solve it have been posted there yet. It looks very alluring, so I decided to repost it here:
Prove:
\begin{align}
&\int_0^1\ln(1-...
- 5,352
33
votes
7
answers
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Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$
I am trying to prove that
\begin{equation}
\int_{0}^{1}\frac{\log\left(x\right)
\log\left(\,{1 - x^{4}}\,\right)}{1 + x^{2}}
\,\mathrm{d}x = \frac{\pi^{3}}{16} - 3\mathrm{G}\log\left(2\right)
\tag{1}
\...
- 11.7k
34
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9
answers
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Evaluating $\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$
In this thread
a friend posted the following integral
$$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{\...
- 14.6k