All Questions
45
questions
3
votes
0
answers
92
views
What is $ \int_{0}^{\exp(-1)} \frac{\ln \ln \frac{1}{x}}{1+x^{2}} dx $?
Background
According to p. 22 of the following paper by Blagouchine, we have the following Malmsten integral evaluation: $$ \int_{0}^{1} \frac{\ln \ln \frac{1}{x}}{1+x^{2}} dx = \frac{\pi}{2} \ln\left(...
- 7,148
3
votes
1
answer
146
views
how to evaluate this integral $\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$
Question statement: how to evaluate this integral $$\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$$
I don't know if there is a closed form for this integral or not.
Here is my attempt to solve ...
3
votes
0
answers
129
views
evaluate $\int_{0}^{1} \log\left(\frac{(\sqrt{1 - x} - 1)(\sqrt{1 - x^2} + 1)}{(\sqrt{1 - x} + 1)(\sqrt{1 - x^2} - 1)}\right) \arcsin(x^2) \, dx$
$$\int_{0}^{1} \log\left(\frac{(\sqrt{1 - x} - 1)(\sqrt{1 - x^2} + 1)}{(\sqrt{1 - x} + 1)(\sqrt{1 - x^2} - 1)}\right) \arcsin(x^2) \, dx = \frac{\pi^3}{12} - \pi\left(6\sqrt{2} + 2 + \log\left(\frac{3}...
8
votes
5
answers
611
views
Evaluation of $\int_0^1\frac{\log x\,dx}{\sqrt{x(1-x)(1-cx)}}$
Assume $c$ is a small real number.
QUESTION. What is the value of this integral in terms of the complete elliptic function $K(k)$?
$$\int_0^1\frac{\log x}{\sqrt{x(1-x)(1-cx)}}\,dx.$$
I got as far as ...
- 339
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^...
- 2,702
2
votes
1
answer
171
views
Closed form for $\int_0^1 e^{\frac{1}{\ln(x)}}dx$?
I want to evaluate and find a closed form for this definite integral:$$\int_0^1 e^{\frac{1}{\ln(x)}}dx.$$
I don't know where to start. I've tried taking the natural logarithm of the integral, ...
- 756
6
votes
1
answer
296
views
An integration $ \int_0^1 x\ln \left ( \sqrt{1+x}+\sqrt{1-x}\right)\ln \left ( \sqrt {1+x} -\sqrt{1-x} \right)\mathrm{d}x.$
How can we evaluate
$$\int_0^1 x\ln \left ( \sqrt{1+x}+\sqrt{1-x}\right)\ln \left ( \sqrt {1+x} -\sqrt{1-x} \right)\mathrm{d}x?$$
Usually when having as the integrand a logarithmic function, the ...
- 201
23
votes
2
answers
1k
views
What is $\int_0^1 \ln (1-x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx$?
There are well-known closed-form evaluations for integrals of the form $\int_0^1 a(x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx $ for certain algebraic functions $a(x)$. For example, an ...
- 2,017
4
votes
1
answer
405
views
Summation of series containing logarithm: $\sum_{n=1}^\infty \ln \frac{(n+1)(3n+1)}{n(3n+4)}$
How do I find the sum of the series: $$\ln \frac{1}{4} + \sum_{n=1}^\infty \ln \frac{(n+1)(3n+1)}{n(3n+4)} $$
I tried expanding the terms on numerator and denominator and got $$\ln \frac{1}{4} + \...
- 355
21
votes
1
answer
578
views
Evaluating $\sum_{n \geq 1}\ln \!\left(1+\frac1{2n}\right) \!\ln\!\left(1+\frac1{2n+1}\right)$
Is there a direct way to evaluate the following series?
$$
\sum_{n=1}^{\infty}\ln \!\left(1+\frac1{2n}\right) \!\ln\!\left(1+\frac1{2n+1}\right)=\frac12\ln^2 2. \tag1
$$
I've tried telescoping ...
- 121k
9
votes
3
answers
254
views
Closed-form of $\int_0^1 x^n \operatorname{li}(x^m)\,dx$
I've conjectured, that for $n\geq0$ and $m\geq1$ integers
$$
\int_0^1 x^n \operatorname{li}(x^m)\,dx \stackrel{?}{=} -\frac{1}{n+1}\ln\left(\frac{m+n+1}{m}\right),
$$
where $\operatorname{li}$ is the ...
- 12.4k
2
votes
1
answer
327
views
Solving $x - a \log(x)=b$
Let $a>0$ and $b \in \mathbb{R}$: Assume there exists an $x >0 $ s.t.
$$x - a\log(x) = b$$
holds. How can it be determined in closed-form?
- 990
2
votes
2
answers
100
views
HowTo solve this integral involving logarithm
I would like to solve integrals of the form
$$I(c) := \int_0^\infty \log(1+x) x^{-c} \, dx ,$$
where $c \in (1,2)$.
Mathematica says either
1) $I(c) = \frac{\pi}{1-c} \csc(\pi c)$
or
2) $I(c) = \...
- 990
2
votes
3
answers
92
views
Prove that for any positive integer $n$ and $d$, $\sum_{k=0}^d 2^k\log_2(\frac{n}{2^k})=2^{d+1}\log_2(\frac{n}{2^{d-1}})-2-\log_2{n}$
I could prove it by induction, but I need to see how I might have discovered it for myself (cause that's what's gonna be on exam).
- 813
6
votes
3
answers
387
views
How to solve $\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$?
Here is my question
$$\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$$
I have tried it by substituting $x$ = $\frac{1}{t}$. I got the answer $0$ but the correct answer is $\pi log(2)$. Any ...
- 2,586