All Questions
18
questions
6
votes
1
answer
285
views
Calculate $\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$
this integral got posted on a mathematics group by a friend
$$I=\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$$
I tried seeing what I'd get from ...
- 231
3
votes
0
answers
316
views
Two tough integrals with logarithms and polylogarithms
The following two integrals are given in (Almost) Impossible Integrals, Sums, and Series (see Sect. $\textbf{1.55}$, page $35$),
$$i) \int_0^{\pi/2} \cot (x) \log (\cos (x)) \log ^2(\sin (x)) \...
- 5,495
3
votes
1
answer
314
views
Evaluating $\int_0^1\frac{\ln^2(1+x)+2\ln(x)\ln(1+x^2)}{1+x^2}dx$
How to show that
$$\int_0^1\frac{\ln^2(1+x)+2\ln(x)\ln(1+x^2)}{1+x^2}dx=\frac{5\pi^3}{64}+\frac{\pi}{16}\ln^2(2)-4\,\text{G}\ln(2)$$
without breaking up the integrand since we already know:
$$\int_0^1\...
- 25.8k
12
votes
2
answers
718
views
General expressions for $\mathcal{L}(n)=\int_{0}^{\infty}\operatorname{Ci}(x)^n\text{d}x$
Define $$\operatorname{Ci}(x)=-\int_{x}^{
\infty} \frac{\cos(y)}{y}\text{d}y.$$
It is easy to show
$$
\mathcal{L}(1)=\int_{0}^{\infty}\operatorname{Ci}(x)\text{d}x=0
$$
and
$$\mathcal{L}(2)=\int_{0}^{\...
- 5,370
6
votes
1
answer
494
views
Is the closed form of $\int_0^1 \frac{x\ln^a(1+x)}{1+x^2}dx$ known in the literature?
We know how hard these integrals
$$\int_0^1 \frac{x\ln(1+x)}{1+x^2}dx;
\int_0^1 \frac{x\ln^2(1+x)}{1+x^2}dx;
\int_0^1 \frac{x\ln^3(1+x)}{1+x^2}dx;
...$$
can be. So I decided to come up with a ...
- 25.8k
23
votes
3
answers
2k
views
Challenging problem: Calculate $\int_0^{2\pi}x^2 \cos(x)\operatorname{Li}_2(\cos(x))dx$
The following problem is proposed by a friend:
$$\int_0^{2\pi}x^2 \cos(x)\operatorname{Li}_2(\cos(x))dx$$
$$=\frac{9}{8}\pi^4-2\pi^3-2\pi^2-8\ln(2)\pi-\frac12\ln^2(2)\pi^2+8\ln(2)\pi G+16\pi\Im\left\{\...
- 25.8k
8
votes
2
answers
270
views
Is there a closed form for $\int_0^1 \binom{1}{x}\frac{\log^2(1-x)}{x}\ \mathrm{d}x$?
Do we know if there is a closed form for
$$
I :=\int_0^1 \binom{1}{x}\frac{\log^2(1-x)}{x}\ \mathrm{d}x\mathrm{?}
$$
Wolfram alpha gives an approximation of $2.66989$ which may be equivalent to:
$$10\...
user753116
5
votes
2
answers
452
views
Calculating $\int_0^1\frac{1}{1+x}\operatorname{Li}_2\left(\frac{2x}{1+x^2}\right)dx$
How to prove in a simpe way that
$$\int_0^1\frac{1}{1+x}\operatorname{Li}_2\left(\frac{2x}{1+x^2}\right)dx=\frac{13}{8}\ln2\zeta(2)-\frac{33}{32}\zeta(3)\ ?$$
where $\operatorname{Li}_2$ is the ...
- 25.8k
8
votes
0
answers
404
views
Powerful Integral $\int_0^x\frac{t\ln(1-t)}{1+t^2}\ dt$
This integral can be found in Cornel's book, (Almost) Impossible Integral, Sums and Series page $97$ where he showed that
$$\int_0^x\frac{t\ln(1-t)}{1+t^2}\ dt=\frac14\left(\frac12\ln^2(1+x^2)-2\...
- 25.8k
4
votes
3
answers
686
views
Evaluate $\int\limits_0^\infty \frac{\ln^2(1+x)}{1+x^2}\ dx$
This problem was already solved here (in different closed form).
But how can you prove $\ \displaystyle\int\limits_0^\infty\frac{\ln^2(1+x)}{1+x^2}\ dx=2\Im\left(\operatorname{Li}_3(1+i)\right)\ $
...
- 25.8k
2
votes
0
answers
368
views
Upper bound the Polylogarithm $\sum_{n=1}^\infty \frac{x^n}{n^2}$
Let $x \in (0,1)$ be some real number, we can then consider the Polylogarithm:
$$\operatorname{L}_2(x)=\sum_{n=1}^\infty \frac{x^n}{n^2}$$
It is not hard to see that the following upper bound holds:
$$...
- 1,135
9
votes
1
answer
218
views
On the monotony of $-\int_0^1\frac{e^y}{y}(\operatorname{Li}_x(1-y)-\zeta(x)) \,\mathrm dy$
$\def\d{\mathrm{d}}$After I was studying variations of the integral representation for $\zeta(3)$ due to Beukers, see the section More complicated formulas from this Wikipedia, I've thought an ...
user243301
11
votes
3
answers
561
views
Is the following Harmonic Number Identity true?
Is the following identity true?
$$ \sum_{n=1}^\infty \frac{H_nx^n}{n^3} = \frac12\zeta(3)\ln x-\frac18\ln^2x\ln^2(1-x)+\frac12\ln x\left[\sum_{n=1}^\infty\frac{H_{n} x^{n}}{n^2}-\operatorname{Li}_3(...
- 5,558
3
votes
1
answer
513
views
Another way of doing integration
What's your option for calculating this integral? No full solution is necessary, it's optional as usual.
Calculate
$$\int_0^1 \frac{2 \zeta (3)\log ^3(1-x) \text{Li}_2(1-x) }{x}-\frac{2 \zeta (3) \...
- 44.4k
19
votes
3
answers
709
views
Calculating in closed form $\int_0^{\pi/2} \arctan\left(\sin ^3(x)\right) \, dx \ ?$
It's not hard to see that for powers like $1,2$, we have a nice closed form. What can be said about
the cubic version, that is
$$\int_0^{\pi/2} \arctan\left(\sin ^3(x)\right) \, dx \ ?$$
What are ...
- 44.4k