All Questions
Tagged with logarithms definite-integrals
298
questions
4
votes
2
answers
287
views
Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$
While exploring possible applications for exponential substitution, I stumbled upon the following integral identity:
$$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
- 1,011
3
votes
1
answer
62
views
Euler Sums of Weight 6
For the past couple of days I have been looking at Euler Sums, and I happened upon this particular one:
$$
\sum_{n=1}^{\infty}\left(-1\right)^{n}\,
\frac{H_{n}}{n^{5}}
$$
I think most people realize ...
- 203
8
votes
1
answer
177
views
how to integrate $\int_0^1 \ln^4(1+x) \ln(1-x) \, dx$?
I'm trying to evaluate the integral $$\int_0^1 \ln^4(1+x) \ln(1-x) \, dx,$$ and I'd like some help with my approach and figuring out the remaining steps.
or is it possible to evaluate $$\int_0^1 \ln^n(...
-1
votes
4
answers
97
views
Calculate $\lim_{n\to \infty}\int_n^{n+1}\frac{1-\ln{x}}xdx$ [duplicate]
$$
\mbox{Given}\ \operatorname{F}\left(x\right) \equiv
\int_{1}^{x}\frac{1-\ln\left(t\right)}{t}\,{\rm d}t,\ \mbox{calculate}\ \lim_{n \to \infty}\left[\operatorname{F}\left(n + 1\right) - \...
- 345
3
votes
1
answer
146
views
how to evaluate this integral $\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$
Question statement: how to evaluate this integral $$\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$$
I don't know if there is a closed form for this integral or not.
Here is my attempt to solve ...
6
votes
0
answers
156
views
Finding a closed form for $ \int_0^1 \frac1x \ln\left(\frac{\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x $
I want a closed form for the following integral
$$
\int_0^1 \frac1x\;\ln\left(\frac {\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x
$$
An integration by parts ...
- 203
1
vote
2
answers
123
views
How to integrate $\int_{0}^{1} \frac{x \operatorname{Li}_2(1 - x)}{1 + x^2} \, dx$
How to integrate $$\int_{0}^{1} \frac{x \operatorname{Li}_2(1 - x)}{1 + x^2} \, dx$$
My try to integrate
$$\text{I}=\int_{0}^{1} \frac{x \operatorname{Li}_2(1 - x)}{1 + x^2} \, dx$$
\begin{aligned}
&...
4
votes
1
answer
104
views
Tricky integral involving Arctan
I'm trying to do an integral which arises from another integral that is simple in polar coordinates (so I can work out the answer that way). However I am using cartesian coordinates and trying to see ...
- 53
4
votes
1
answer
222
views
Evaluation of $\int_{0}^{\infty} \, \frac{x^\alpha \log^n{(x^\beta)}}{k^m+x^m}\, dx$
How to evaluate the following integral/
$$\int_{0}^{\infty} \, \frac{x^\alpha \log^n{(x^\beta)}}{k^m+x^m}\, dx $$, $\alpha, \beta, m, n$ are real and $k$ is a positive integer.
For the case where $\...
- 107
5
votes
3
answers
161
views
About the integral $\int_{0}^{2\pi} \ln(s^2x^2 + 1) \, ds $
I've encountered an integral that has piqued my interest and am seeking insights or methods for its evaluation. The integral in question is:
$$
\int_{0}^{2\pi} \ln(s^2x^2 + 1) \, ds
$$
where ($x$) is ...
- 311
2
votes
2
answers
105
views
Evaluate: $\int_{-2}^{2} \frac{x^2}{1+5^{x}} dx$
Evaluate:
$$\int_{-2}^{2} \frac{x^2}{1+5^{x}} dx$$
After checking f(-x) for odd/even function and not getting suitable results.
Let $$1+5^x=t$$
$$dx=\frac{dt}{log(5)(t-1)}$$
$$x^2=\frac{(t-1)^2}{(log(...
- 155
3
votes
0
answers
129
views
evaluate $\int_{0}^{1} \log\left(\frac{(\sqrt{1 - x} - 1)(\sqrt{1 - x^2} + 1)}{(\sqrt{1 - x} + 1)(\sqrt{1 - x^2} - 1)}\right) \arcsin(x^2) \, dx$
$$\int_{0}^{1} \log\left(\frac{(\sqrt{1 - x} - 1)(\sqrt{1 - x^2} + 1)}{(\sqrt{1 - x} + 1)(\sqrt{1 - x^2} - 1)}\right) \arcsin(x^2) \, dx = \frac{\pi^3}{12} - \pi\left(6\sqrt{2} + 2 + \log\left(\frac{3}...
12
votes
2
answers
242
views
Does $\int_{1}^{2}\ln\left(x+\ln\left(x^2+\ln\left(x^3+\ln\left(\dots\right)\right)\right)\right)dx=1$?
I was messing around with the infinitely nested logarithm $f(x)=\ln\left(x+\ln\left(x^2+\ln\left(x^3+\ln\left(\dots\right)\right)\right)\right)$ on Desmos when I decided to take the integral from $x = ...
- 1,688
7
votes
2
answers
761
views
Evaluation of $\int_0^1\frac{\ln ^2\left( 1-x \right) \text{Li}_2\left( \dfrac{1+x}{2} \right)}{x}\text{d}x$
show that
\begin{align*} \int_0^1\frac{\ln ^2\left( 1-x \right) \text{Li}_2\left( \dfrac{1+x}{2} \right)}{x}\text{d}x&=\text{2Li}_5\left( \frac{1}{2} \right) +\text{2}\ln\text{2Li}_4\left( \frac{...
user1235604
2
votes
2
answers
144
views
Is there a closed form solution to $\sum_{n=1}^{\infty}\frac{\ln\left(\frac{n+1}{n}\right)}{n}$ [duplicate]
While coming up with an idea for another way to milk the integral in my previous question, I got stuck at this summation: $$\sum_{n=1}^{\infty}\frac{\ln\left(\frac{n+1}{n}\right)}{n}$$
I do not know ...
- 1,688