All Questions
73
questions
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How to integrate $\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$ [duplicate]
Question; how to integrate $$\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$$
here is my attempt to solve the integral
\begin{align} I&=\int_0^\frac{1}{2}\frac{\ln(1+...
1
vote
0
answers
51
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Polylogarithmically solving $\int\frac{\log(a_1x+b_1)\cdots\log(a_nx+b_n)}{px+q}\,dx$
I am now trying a direct approach to solving my question about
$$\int_0^\infty\frac{\arctan a_1x\arctan a_2x\dots\arctan a_nx}{1+x^2}\,dx$$
where the $a_i$ are all positive. Note that the $\arctan$s ...
- 104k
8
votes
1
answer
285
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Evaluate $\int_0^\infty\frac{dx}{1+x^2}\prod_i\arctan a_ix$ (product of arctangents and Lorentzian)
Define
$$I(a_1,\dots,a_n)=\int_0^\infty\frac{dx}{1+x^2}\prod_{i=1}^n\arctan a_ix$$
with $a_i>0$. By this answer $\newcommand{Li}{\operatorname{Li}_2}$
$$I(a,b)=
\frac\pi4\left(\frac{\pi^2}6
-\Li\...
- 104k
3
votes
0
answers
186
views
how to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?
here in my answer I got real part for polylogarithm function at $1+i$ for natural $n$
$$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$
where
$$ B_n=\sum_{k=0}^{\lfloor\...
- 1,637
4
votes
0
answers
112
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Calculate an integral involving polylog functions
Im my recent answer https://math.stackexchange.com/a/4777055/198592 I found numerically that the following integral has a very simple result
$$i = \int_0^1 \frac{\text{Li}_2\left(\frac{i\; t}{\sqrt{1-...
- 12.7k
5
votes
1
answer
193
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Evaluating $\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{(ax)}\operatorname{arsinh}{(bx)}}{x}$ in terms of polylogarithms
Define the function $\mathcal{I}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ by the definite integral
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{\left(ax\right)}\...
- 30.7k
5
votes
1
answer
288
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Closed forms of the integral $ \int_0^1 \frac{\mathrm{Li}_n(x)}{(1+x)^n} d x $
(This is related to this question).
How would one find the closed forms the integral
$$ \int_0^1 \frac{\mathrm{Li}_n(x)}{(1+x)^n} d x?
$$
I tried using Nielsen Generalized Polylogarithm as mentioned ...
- 107
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
3
votes
1
answer
215
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Is there an analytic solution to $\int_a^b \frac{\arctan(A+Bt)}{C^2 + (t-Z)^2}dt$?
Is there a sensible analytic solution to the following integral: $$\int_a^b \frac{\arctan(A+Bt)}{C^2 + (t-Z)^2}dt$$ where all constants are real and $C>0$.
This integral is part of the third term ...
- 31
10
votes
1
answer
790
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A generalized "Rare" integral involving $\operatorname{Li}_3$
In my previous post, it can be shown that
$$\int_{0}^{1}
\frac{\operatorname{Li}_2(-x)-
\operatorname{Li}_2(1-x)+\ln(x)\ln(1+x)+\pi x\ln(1+x)
-\pi x\ln(x)}{1+x^2}\frac{\text{d}x}{\sqrt{1-x^2} }
=\...
- 5,370
17
votes
1
answer
1k
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A rare integral involving $\operatorname{Li}_2$
A rare but interesting integral problem:
$$\int_{0}^{1}
\frac{\operatorname{Li}_2(-x)-
\operatorname{Li}_2(1-x)+\ln(x)\ln(1+x)+\pi x\ln(1+x)
-\pi x\ln(x)}{1+x^2}\frac{\text{d}x}{\sqrt{1-x^2} }
=\...
- 5,370
8
votes
2
answers
494
views
Finding $\int_{1}^{\infty} \frac{1}{1+x^2} \frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{\pi^2+\ln^2\left(\frac{x-1}{2}\right)}\text{d}x$
Prove the integral
$$\int_{1}^{\infty} \frac{1}{1+x^2}
\frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{
\pi^2+\ln^2\left ( \frac{x-1}{2} \right ) }\text{d}x
=\frac{96C\ln2+7\pi^3}{12(\pi^2+...
- 5,370
2
votes
1
answer
126
views
Closed form evaluation of a class of inverse hyperbolic integrals
Define the function $\mathcal{I}:\mathbb{R}_{>0}^{2}\rightarrow\mathbb{R}$ via the definite integral
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{\left(ax\...
- 30.7k
7
votes
1
answer
196
views
Iterated integral involving polylogarithms
To establish notation the polylogarithm Li$_n(x)$ has the power series expansion
$$ \text{Li}_n(x)= \sum_{k=1}^\infty \frac{x^k}{k^n} $$
and the Riemann zeta can be considered the special value $\zeta(...
- 6,760
3
votes
1
answer
368
views
Challenging integral $I=\int_0^{\pi/2}x^2\frac{\ln(\sin x)}{\cos x}dx$
My friend offered to solve this integral.
$$I=\int_0^{\pi/2}x^2\frac{\ln(\sin x)}{\cos x}dx=\frac{\pi^4}{32}-{4G^2} $$
Where G is the Catalan's constant.
$$I=\int _0^{\infty }\frac{\arctan ^2\left(u\...
- 5,507