Linked Questions

408
votes
8answers
196k views

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$

I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The ...
142
votes
3answers
30k views

Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$

I am trying to find a closed form for $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$ It seems that the answer is $$\frac{\pi^2}{12}\left( 1-\...
27
votes
5answers
1k views

Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$

Today I discussed the following integral in the chat room $$\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$$ where $0\leq a, b\leq \pi$ and $k&...
3
votes
5answers
3k views

How many infinite series representations of the golden ratio are in existence?

How many infinite series representations of the golden ratio are in existence? All I can find is one that expands out the $5^{1/2}$ part in $\varphi= \frac12(1+5^{1/2})$ and the one that uses the ...
8
votes
1answer
880 views

Evaluating $\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$ using complex analysis

how do I compute $$\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$$ with complex analysis? I feel like im calculating the residue wrong and I cant get to the answer correctly. I tried to branch cut ...
4
votes
1answer
1k views

golden ratio from new formula? perhaps from theory of modular units?

Please consider the following infinite product series which I found by pure happenstance: $$\frac{1+\sqrt{5}}{2}= e^{\pi/6} \prod_{k=1}^\infty \frac{1+e^{-5(2k-1)\pi}}{1+e^{-(2k-1)\pi}}$$ My ...
10
votes
1answer
322 views

Nontrivial integral representations for $e$

There are a lot of integral representations for $\pi$ as well as infinite series, limits, etc. For other transcendental constants as well (like $\gamma$ or $\zeta(3)$). However, for every definite ...
7
votes
2answers
213 views

Conjecture for the value of $\int_0^1 \frac{1}{1+x^{p}}dx$

While browsing the post Is there any integral for the golden ratio $\phi$?, I came across this nice answer, $$ \int_0^\infty \frac{1}{1+x^{10}}dx=\frac{\pi\,\phi}5$$ it seems the general form is just ...
7
votes
1answer
152 views

Other integrals for the tribonacci constant?

The post, Is there an integral for the golden ratio? gives numerous beautiful integrals for $\phi$. Some were just specializations of trigonometric evaluations such as, $$F(k)=\int_0^\infty \frac{x^{\...
11
votes
0answers
390 views

The most complex formula for the golden ratio $\varphi$ that I have ever seen. How was it achieved?

I am fascinated by the following formula for the golden ratio $\varphi$: $$\Large\varphi = \frac{\sqrt{5}}{1 + \left(5^{3/4}\left(\frac{\sqrt{5} - 1}{2}\right)^{5/2} - 1\right)^{1/5}} - \frac{1}{e^{2\...
2
votes
1answer
88 views

On the integral $\int_0^1 \frac{dx}{\sqrt[3]{x+\sqrt[3]{x+\sqrt[3]{x+\cdots}}}}$ and the plastic constant

We have, $$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$$ $$P=\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\cdots}}}$$ with golden ratio $\phi$ and plastic constant $P$. If, $$\int_0^1 \frac{dx}{\sqrt{x+\sqrt{x+\...
1
vote
0answers
93 views

Elementary solution of $\int_0^1\frac{\ln(1-3x+x^2)\ln x}xdx$

Is there a elementary way to evaluate $$\int_0^1\frac{\ln(1-3x+x^2)\ln x}xdx=\frac85\zeta(3)+\frac{2}{5} \pi ^2 \ln\varphi-2 i \pi\ln^2\varphi$$ where $\varphi$ is the golden ratio? (I posted it ...