Linked Questions

22
votes
7answers
2k views

Integral: $\int_0^{\pi/12} \ln(\tan x)\,dx$

I am trying to evaluate: $$\int_0^{\pi/12} \ln(\tan x)\,dx$$ I think the integral is quite simple but I am having a hard time evaluating it. I started with the result: $$\int_0^{\pi/4} \ln(\tan x)\,...
12
votes
5answers
348 views

Seeking Methods to solve $ I = \int_{0}^{\frac{\pi}{2}} \frac{\arctan\left(\sin(x)\right)}{\sin(x)}\:dx$

I was wondering what methods people knew of to solve the following definite integral? I have found a method using Feynman's Trick (see below) but am curious as to whether there are other Feynman's ...
10
votes
6answers
714 views

How to show that $ \int^{\infty}_{0} \frac{\ln (1+x)}{x(x^2+1)} \ dx = \frac{5{\pi}^2}{48} $ without complex analysis?

The Problem I am trying to show that $ \displaystyle \int^{\infty}_{0} \frac{\ln (1+x)}{x(x^2+1)} \ dx = \frac{5{\pi}^2}{48}$ My attempt I've tried substituting $x=\tan\theta$, and then using the ...
6
votes
6answers
515 views

Request for crazy integrals

I'm a sucker for exotic integrals like the one evaluated in this post. I don't really know why, but I just can't get enough of the amazing closed forms that some are able to come up with. So, what ...
12
votes
3answers
472 views

Baaad: $\int_0^{\pi/2}\log\bigg(\frac{2+\sin2x}{2-\sin2x}\bigg)\mathrm dx$

I am in the process of proving $$I=\int_0^\infty \frac{\arctan x}{x^4+x^2+1}\mathrm{d}x=\frac{\pi^2}{8\sqrt{3}}-\frac23G+\frac\pi{12}\log(2+\sqrt{3})$$ And I have gotten as far as showing that $$2I=\...
12
votes
4answers
335 views

Integral $\int_0^\infty \frac{\ln(1+x+x^2)}{1+x^2}dx$

Prove that$$I=\int_0^\infty \frac{\ln(1+x+x^2)}{1+x^2}dx=\frac{\pi}{3}\ln(2+\sqrt 3)+\frac43G$$ I've found this integral in my notebook and perhaps I encountered it before since it looks quite ...
9
votes
3answers
322 views

Integral $\int_0^\frac{\pi}{2} \arcsin(\sqrt{\sin x}) dx$

I am trying to calculate $$I=\int_0^\frac{\pi}{2} \arcsin(\sqrt{\sin x}) dx$$ So far I have done the following. First I tried to let $\sin x= t^2$ then: $$I=2\int_0^1 \frac{x\arcsin x}{\sqrt{1-x^4}}...
2
votes
4answers
158 views

Seeking methods to solve $\int_{0}^{\infty} \frac{x - \sin(x)}{x^3\left(x^2 + 4\right)} \:dx$

I recently asked for definite integrals that can be solved using the Feynman Trick. One of the responses is the following integral: $$I = \int_{0}^{\infty} \frac{x - \sin(x)}{x^3\left(x^2 + 4\right)} ...
4
votes
2answers
139 views

Seeking methods to solve $ \int_{0}^{\frac{\pi}{2}} \ln\left|2 + \tan^2(x) \right| \:dx $

As part of going through a set of definite integrals that are solvable using the Feynman Trick, I am now solving the following: $$ \int_{0}^{\frac{\pi}{2}} \ln\left|2 + \tan^2(x) \right| \:dx $$ I'...
3
votes
2answers
134 views

What methods can be used to solve $ \int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx $

I'm seeking methods to solve the following definite integral: $$ I = \int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx $$
6
votes
1answer
222 views

Evaluate $\int_0^{\infty} \frac {\ln(1+x^3)}{1+x^2}dx$

Prove that $$\int_0^{\infty} \frac {\ln(1+x^3)}{1+x^2}dx=\frac {\pi \ln 2}{4}-\frac {G}{3}+\frac {2\pi}{3}\ln(2+\sqrt 3)$$ Where $G$ is the Catalan's constant. Actually I proved this using the ...
9
votes
2answers
188 views

Seeking Methods to solve $\int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx $

After weeks of going back and forth I've been able to solve the following definite integral: $$I = \int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx $$ To solve this I employ ...
3
votes
2answers
140 views

$\int_0^\frac{\pi}{2}\frac{\ln(\sin(x))\ln(\cos(x))}{\tan(x)}dx$

I have the problem below: $$\int_0^\frac{\pi}{2}\frac{\ln(\sin(x))\ln(\cos(x))}{\tan(x)}dx$$ I have tried $u=\ln(\sin(x))$ so $dx=\tan(x)du$ so the integral becomes: $$\int_{-\infty}^0u\ln(\cos(x))du$...
5
votes
1answer
117 views

Integral $\int_0^\infty \frac{x-\sin x}{x^3(x^2+4)} dx$

The following integral appeared on the $8$th Open Mathematical Olympiad of the Belarusian-Russian University. $$I=\int_0^\infty \frac{x-\sin x}{x^3(x^2+4)} dx$$ I used power series: $$x-\sin x = \sum_{...
9
votes
1answer
224 views

Using Laplace transforms to evaluate$\int_{0}^{\infty}\frac{\sin^2(x)}{x^2(x^2 + 1)} dx$

Recently I've been playing around with Feynman's Trick to evaluate integrals. Obviously, one of it's many great features is that it allows derivatives to make expressions simpler. I was wondering ...

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