All Questions
13
questions
4
votes
1
answer
257
views
Find closed-form of: $\int_{0}^{1}\frac{x\log^{3}{(x+1)}}{x^2+1}dx$
I found this integral: $$\int_{0}^{1}\frac{x\log^{3}{(x+1)}}{x^2+1}dx$$
And it seems look like this problem but i don't know how to process with this one.
First, i tried to use series of $\frac{x}{x^...
- 2,732
1
vote
1
answer
83
views
Integral of a modified softplus function
In a manuscript I am currently reading, the authors propose a modified softplus function
$$g(a)=\frac{\log\left(2^a +1 \right)}{\log(2)}$$
for some $a \in \mathbb{R}$. The authors then claim that if $...
- 961
1
vote
0
answers
74
views
Integrate $\int_{-\infty}^\infty [4(\log r_1 - \log r_2) - 2(x_1^2/r_1^2 - x_2^2/r_2^2)]^2 dx$
As the title suggests, I am having trouble evaluating the following definite integral:
$$\int_{-\infty}^\infty \left[4\left(\log r_1 - \log r_2\right) - 2\left(\frac{x_1^2}{r_1^2} - \frac{x_2^2}{r_2^...
- 11
0
votes
0
answers
135
views
Validity of argument in dilogarithm identities on Wolfram
I've come across a series of identities existing between dilogarithms and powers of logarithms but I am not sure about when such equations are valid in terms of the restriction of the domain of the ...
- 2,850
1
vote
2
answers
495
views
How to evaluate the integral $\int_0^2 \frac{\ln x}{\sqrt {x^2-2x+2}}dx$?
Yesterday's integral may be too difficult, I think the following integral should not be difficult.
$$I=\int_0^2 \frac{\ln x}{\sqrt {x^2-2x+2}}dx=\int_{-1}^1\frac{\ln(x+1)}{\sqrt {x^2+1}}dx=\int_{-1}^1\...
- 1,431
-1
votes
1
answer
199
views
intersection between exponential and polylogarithmic functions
It's possible to solve this equation without using Lambert function or any numerical method, but only with ordinary algebra?
$n^{k}lg_2(n) \le k^n$ with $k,n>0, k \in \mathbb{R}$
For $k=\frac{4}{...
- 23
0
votes
2
answers
150
views
Complex logarithms when computing real-valued integral
My question arise when I try to calculate real-valued integral, specifically, I want to evaluate the integral
\begin{equation} \int_0^1 \frac{\ln \left(\frac{x^2}{2}-x+1\right)}{x} dx
\end{equation}
...
- 138
8
votes
3
answers
342
views
Integral $\int_0^{1/2}\arcsin x\cdot\ln^3x\,dx$
It's a follow-up to my previous question.
Can we find an anti-derivative
$$\int\arcsin x\cdot\ln^3x\,dx$$
or, at least, evaluate the definite integral
$$\int_0^{1/2}\arcsin x\cdot\ln^3x\,dx$$
in a ...
- 1,928
16
votes
1
answer
551
views
Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points
The polylogarithm is defined by the series
$$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$
There are relations connecting values of the polylogarithm at certain rational points in the ...
- 47.3k
15
votes
3
answers
2k
views
Simplification of an expression containing $\operatorname{Li}_3(x)$ terms
In my computations I ended up with this result:
$$\mathcal{K}=78\operatorname{Li}_3\left(\frac13\right)+15\operatorname{Li}_3\left(\frac23\right)-64\operatorname{Li}_3\left(\frac15\right)-102
\...
- 5,342
21
votes
1
answer
921
views
Closed form for ${\large\int}_0^1\frac{\ln^3x}{\sqrt{x^2-x+1}}dx$
This is a follow-up to my earlier question Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$.
Is there a closed form for this integral?
$$I=\int_0^1\frac{\ln^3x}{\sqrt{x^2-x+1}}dx\...
- 47.3k
41
votes
2
answers
2k
views
Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$
Here is another integral I'm trying to evaluate:
$$I=\int_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx.\tag1$$
A numeric approximation is:
$$I\approx-0....
- 47.3k
33
votes
4
answers
3k
views
Integral ${\large\int}_0^1\ln(1-x)\ln(1+x)\ln^2x\,dx$
This problem was posted at I&S a week ago, and no attempts to solve it have been posted there yet. It looks very alluring, so I decided to repost it here:
Prove:
$$\int_0^1\ln(1-x)\ln(1+x)\ln^...
- 5,342