Linked Questions

3 votes
1 answer
238 views

What's the common theme to look out for, while applying Feynman's Integral tricks? [duplicate]

I have seen Feynman's integral trick coming into use in many questions, but I don't really see a common way to recognize the format for it. Of course when a question comes with an explicit parameter, ...
Arsenic's user avatar
  • 321
30 votes
8 answers
5k 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)\,...
Pranav Arora's user avatar
30 votes
10 answers
6k 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 ...
clathratus's user avatar
  • 17.1k
25 votes
3 answers
3k views

Efficiently evaluating $\int x^{4}e^{-x}dx$ [duplicate]

The integral I am trying to compute is this: $$\int x^{4}e^{-x}dx$$ I got the right answer but I had to integrate by parts multiple times. Only thing is it took a long time to do the computations. I ...
user262291's user avatar
  • 1,449
13 votes
5 answers
1k 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 ...
user avatar
13 votes
6 answers
2k 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 ...
Zacky's user avatar
  • 26.6k
16 votes
5 answers
2k views

Solving the integral $\int_0^{\pi/2}\log\left(\frac{2+\sin2x}{2-\sin2x}\right)\mathrm dx$

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=\...
clathratus's user avatar
  • 17.1k
14 votes
4 answers
821 views

Evaluating $\int_0^\pi x\frac{\sin{\frac{x}{2}} - \cos{\frac{x}{2}}}{\sqrt{\sin{x}}} dx$

How am I supposed to solve the following definite integral? $$ \mathcal{I} = \int_0^\pi x \cdot \frac{\sin{\frac{x}{2}} - \cos{\frac{x}{2}}}{\sqrt{\sin{x}}} dx $$ This definite integral is solved if ...
Rohan Bari's user avatar
6 votes
8 answers
554 views

Evaulate $\int_{-\infty}^{\infty} \frac{\ln(x^2+1)}{x^4+x^2+1}dx$

I'm really stuck on this integral. I want to believe there is a closed form for it but I'm really unable to find it. WFA cannot find one either. I think it may be possible to use Feynman's Trick and ...
Anik Patel's user avatar
10 votes
4 answers
558 views

Proving $\int_0^\infty \log\left (1-2\frac{\cos 2\theta}{x^2}+\frac{1}{x^4} \right)dx =2\pi \sin \theta$

Prove $$\int_0^\infty \log \left(1-2\frac{\cos 2\theta}{x^2}+\frac{1}{x^4} \right)dx =2\pi \sin \theta$$where $\theta\in[0,\pi]$. I've met another similar problem, $$ \int_0^{2\pi} \log(1-2r\cos \...
Chiquita's user avatar
  • 2,930
11 votes
3 answers
1k 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}}...
Zacky's user avatar
  • 26.6k
14 votes
1 answer
988 views

A difficult double integral $\int_{0}^{1}\int_{0}^{1}\frac{x\ln x \ln y }{1-xy}\frac{dxdy}{\ln(xy)}$

How to evaluate $$\int_{0}^{1}\int_{0}^{1}\frac{x\ln x\ln y}{1-xy}\frac{dxdy}{\ln(xy)} ?$$ Any ideas on how to even start with this integral? It seems impossible to me. There's a similar integral ...
user avatar
6 votes
2 answers
1k views

Integral $\int_{0}^{\frac{\pi}4} \ln(\sin{x}+\cos{x}+\sqrt{\sin{2x}})dx$

Prove that $$\int_{0}^{\frac{\pi}4} \ln(\sin{x}+\cos{x}+\sqrt{\sin{2x}})dx =\frac{\pi}{4} \ln2$$ I tried to use King's rule and to scale by $2$ and then to add the integrals, to get product of ...
Grentouce's user avatar
  • 301
5 votes
4 answers
923 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'...
user avatar
7 votes
3 answers
827 views

Integrating $\int_0^1\frac{\ln^2x\ln(1+x)}{1+x^2} dx$ using real methods

How to evaluate, without contour integration the following integral: $$I=\int_0^1\frac{\ln^2x\ln(1+x)}{1+x^2}\ dx\ ?$$ @Cody mentioned in this solution that $$I=\frac{\pi^{2}}{6}G+\frac{\pi^{3}}{...
Ali Shadhar's user avatar
  • 25.2k

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