All Questions
11
questions
2
votes
2
answers
148
views
How to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{{(1 + x) \cdot \log(x) - (1 + y) \cdot \log(y)}}{{x - y}} \cdot (1 + \log(xy)) \,dy \,dx$
Question: How to evaluate this integral $$\int_{0}^{1} \int_{0}^{1} \frac{{(1 + x) \cdot \log(x) - (1 + y) \cdot \log(y)}}{{x - y}} \cdot (1 + \log(xy)) \,dy \,dx$$
My messy try
$$\int_{0}^{1} \int_{...
2
votes
0
answers
81
views
how to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{\ln(1 - xy) \cdot \text{Li}_{4}(1 - x)}{x(1 - x)(1 - xy)} \,dy\,dx$
how to evaluate $$\int_{0}^{1} \int_{0}^{1} \frac{\ln(1 - xy) \cdot \text{Li}_{4}(1 - x)}{x(1 - x)(1 - xy)} \,dy\,dx$$
My attempt
$$ \Omega =\int_{0}^{1} \int_{0}^{1} \frac{\ln(1 - xy) \cdot \text{Li}...
4
votes
2
answers
260
views
How to integrate $\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dx\,dy$
how to integrate $$\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dx\,dy$$
My attempt
$$\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dx\,...
4
votes
1
answer
171
views
How to integrate $\int_{0}^{1} \int_{0}^{1} \ln\left(\frac{1}{\sinh^2(x) + \cosh^2(y)}\right) \,dx\,dy$
How to integrate
$$\int_{0}^{1} \int_{0}^{1} \ln\left(\frac{1}{\sinh^2(x) + \cosh^2(y)}\right) \,dx\,dy$$
My attempt
$$\int_{0}^{1} \int_{0}^{1} \ln\left(\frac{1}{\sinh^2(x) + \cosh^2(y)}\right) \,dx\...
1
vote
0
answers
188
views
Closed form for $A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$
Consider the double sums :
$$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$
$$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$
Is there a closed form for $A_3$ ...
- 16.4k
1
vote
0
answers
42
views
Analytical Solution for a Double Integral Involving Logistic Functions and Gaussian Distributions
I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows:
$$...
- 23
0
votes
0
answers
39
views
Finding a closed-form expression for a definite Integral
Assuming that $a$ and $b$ are positive real numbers, and $c$, $d$, $x_{0}$, and $y_{0}$ are all real. I'm trying to find a closed-form expression for the following definite integral.
\begin{equation}
...
- 33
1
vote
1
answer
59
views
Need to find Closed form solution to interesting problem
$A= (1+x)(1+x-y)^{1/y}$
$B= x(1+x)^{1/y}$
for $y\in(0,1)$ and $x > 0$;
I want to show that A > B, but could not prove it mathematically. I run a simulation and it shows that in fact A > B, but I ...
- 13
3
votes
1
answer
457
views
Evaluating a double integral of a complicated rational function
Define the function $Q:\mathbb{C}^{2}\rightarrow\mathbb{C}$ to be the binary quadratic form,
$$Q{\left(z,w\right)}:=z^{2}+w^{2}.\tag{1a}$$
Also, define $P:\mathbb{C}^{4}\rightarrow\mathbb{C}$ to be ...
- 30.7k
2
votes
1
answer
158
views
A double integral with functions as bounds: $\int_0^1 dx \int_{\frac{\sqrt[3]{x}}2}^{\sqrt[3]{x}} \sqrt{1-y^4}dy$
So I have the integral
$$\int_0^1 dx \int_{\frac{\sqrt[3]{x}}{2}}^{\sqrt[3]{x}} \sqrt{1-y^4}dy.$$
How on earth do I do this integral? I used WolframAlpha to have a look at what $\sqrt{1-y^4}$ ...
- 311
12
votes
2
answers
307
views
Closed form of $\displaystyle\int_{0}^{\pi/4}\int_{\pi/2}^{\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin x)^{y+2}}\, dy\, dx$
Can the following double integral be evaluated analytically
\begin{equation}
I=\int_{0}^{\Large\frac{\pi}{4}}\int_{\Large\frac{\pi}{2}}^{\large\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin x)^{y+2}...
- 19.4k