Doing double Feynman Integration on this hidden Dirichlet Integral ( improper sin^2(x)/x^2 )
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- Опубликовано: 4 янв 2025
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Leibniz Integral rule: • The Leibniz rule for i...
Arctan Integral: • The integral 1/(x^2+1)...
First time Dirichlet: • A COMPLEX BOI! Integra...
Second time Dirichlet: • DESTROYING THE DIRICHL...
Using Laplace transform: • Video
Let us finish this week off with this sweet piece of Integral cake! Never has the evaluation of pi been this good looking :v
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I know that I'm here after 8 months, but I tried it before watching the video and got it right.
My improvement is due to you and Blackpenredpen, mainly.
Thanks.
Same
@@joelsagflaatholmberg3922 same
Dude u absolutely have the best integrals on youtube! Period.
Marko Lazarevic i agree!! I have been learning a lot too!!
I'm so happy I subscribed!!! I was just watching BPRP's 0 to inf version of the Feynman Integral!!! THANK YOU AND KEEP IT UP MAN!
Man you are talented! Keep making these videos, a good integral makes my day
I'm a direct second year entry theoretical physics student at the uni of Edinburgh. I have followed your channel these weeks (months actually) and I found that integration and complex analysis is extremely cool and interesting. In fact, I have already chosen this as my optional course for next year. LOVE THISSS ahah. keep up solving the hard bois. Ciao
4:25 try to find the parameter that works out always makes me so confused
12:04 try to find upper or lower bound by trying to make the other bound vanish is something new to me
nice hat btw
Beautiful video. I think you should have said a>0 when defining I(a) as for a
Great integral!
Great look papa flammy!
The first time i got here was like one year ago and i had no fucking idea of what was going on. Now i got it all right.thanks to You and all of the other maths youtubers. I feel proud of myself lol
Just amazing!
Lol I love it when he says "is nothing else than"
That's bloody amazing
Very nice video :) will spend my sloppy new year's day watching your videos ^^
A quick way is to use integral by part, int ( sin(x)^2/x^2 ) dx = - int (sin(x)^2) d (1/x) = - sin(x)^2/x + int (1/x) d sin(x)^2 the
first part evaluate at 0 and inf give both zero, the second part int (1/x) d sin(x)^2 =int (sin(2x)/x) dx, use u-sub, let u=2x, and use the fact int (sin(u)/u) du = pi/2, we are done!
It is convergible to integral of 2 times sinx/x from 0 to infinity and that would be more easy to solve
Thx it came in my university exam
Keep uploading wonderful problems😊😃
FUDGING BRILLIANT! Is the integration week over?? What else am I supposed to spend my evenings watching????? Back to crippling depression every evening for me :'(
Do you watch anime?
Papa is a man of culture
At 3:55, instead of introducing an exponential function, couldn't you just take the second derivative of i(t) wrt "t" to get rid of the X in the denominator?
Very impressive video. Still amazed by this integral and why it even works
Man this integral is so easy it toke me 2 minuits to solve
Of course I enjoyed, actually I enjoyed a lot (I want to watch it again).
This video is amazing, *COOLIOO* .
Thank you so much *Papa Flammy Mathy Physicy Geppetto Houdini Olivery* .
Hi
Impressive, but I wonder why double Feynman integration is needed.
Having I'(t)=int[0,inf](sin(2tx)/x*dx) if we multiply both numerator and denominator for 2t, standing that d2tx=2tdx, we have I'(t)=int[0,inf](sin(2tx)/2tx*d2tx) that directly evaluates in I'(t)=pi/2 bringing to the final result.
A very good channel with good integration questions
After finding the correct parameter I was able to solve the dirichlet without this video so I'm proud of myself lol
10:54 is it ok to cancel out the a^2, because the condition that we use here is when a=0?
Really enjoyed the video man
There is something really beautiful about using the Leibnitz rule to solve these improper integrals! - in this case might it be more appropriate to use Cauchy's principal value & residue theorem?
wow thanks for this integral ,very awesome question and u look very hilarious,👌👌💂💂
Simply AMAZING !!
Love watching your videos even though it goes completely over my education.
Subscribed straight away.
IRIGIMA V1.80
Why not integrate (sin(x) /x) ^n for all n?
nicely explained
My favorite way to destroy the dirlecht integral!!!
Firstly we observe that integrand is even function
int_{-\infty}^{\infty}\frac{\sin^{2}(x)}{x^2}dx =2int_{0}^{\infty}\frac{\sin^{2}(x)}{x^2}dx
then we use integration by parts
2int_{0}^{\infty}\frac{\sin^{2}(x)}{x^2}dx=-\frac{2sin(x)^2}{x} |_{0}^{\infty} +2\int_{0}^{\infty}{\frac{2\sin{x}\cos{x}}{x}dx}
We calculate limits and simplify integral a little bit
2int_{0}^{\infty}\frac{\sin^{2}(x)}{x^2}dx=4\int_{0}^{\infty}{\frac{sin{(2x)}}{2x}dx}
After this simplification we use substitution
u=2x
du=2dx
2int_{0}^{\infty}\frac{\sin^{2}(x)}{x^2}dx=2\int_{0}^{\infty}{\frac{sin{(u)}}{u}du}
and now Leibniz differentiation under integral sign is optional
;-; how did you get the part in 4:11?
nvm
Could this integral be more easily approached using residue theorem?
there you have it mathhelpboards.com/analysis-50/sin-2x-x-2-complex-analysis-661.html
Gangsta flammy
incredible
You can't go wrong with a solid german dude
Can you do this using complex analysis?
Nice, btw how to solver using the countour intergral?
4:10 - why are you always using the 3 points? I thought they were standing for "therefore".
Yea, right. So you should also use it as one ^^
Is the integral from -inf to inf of any sin(x)^n/x^n when n€N equal to pi?
Fabulous Mannn!! thankiew
Hello papa can you tell me the name of the intro song? Also thanks alot, been trying to integrate this for a while with no succ ess(sorry for bad English)
you should do this:
make a video "integrating" x^x start off with a long intro then just set it equal to T ( some MLG snoop dogg dubstep music starts playing )
is the inverse tangent the same as half a circle? there is alot of pies in there..
Thanks from Turkey...
Can it be solved using series developments? Example: Taylor, Mac Laurin ...
Pls keep doing this kind of video for all Ur lifetime
What is your trick for integrating by parts twice?
i think you can just do it by parts if you integrate 1/x^2 and differentiate sin(x)^2
Sir Please tell me how to solve this problem, If g(x) is the inverse of f(x) and has domain x € [1,5] where f(1)=2 & f(5)= 10 then find the value of integral f(x)dx limit 1 to 5 + g(y)dy limit 2 to 10.????
could i also use e^(-tx), instead of sin(tx) as the new function to get rid of the first x in the denominator and then ofcourse the second time to get rid of the second x?
Could we make a generalisation?
I tried for INTEGRAL[ sin(x^n)/x ] from 0 to inf. and got π/2n. But I couldn't do it in the case of INTEGRAL[(sin x)^n/x^n].
Don't name all functions with a capital I please. :) Besides that, great video, I liked it.
Its traditional to use capital letters for Integrals.
I noticed he defined two different functions, _I(t)_ and _I(a)_ and that confused me!
So cool man!
Excellent.
That was a hell of a week
Wie kann ich dir private Nachrichten schicken? Ich bin da auf etwas interessantes gestoßen
intro song?!!?
Very entertaining
Pi day is coming my bois.
Now who's a good Boi?
this seems dodgy as fuck. At 5:14 we're looking for I'(a=0) but then at 10:32 we're dividing both sides by a^2. And expecting the result to still be valid at a=0. I mean it's probably ok as formally both sides are divided by a parameter that cancels out before we set it to zero, but still I can't help makin a lemon-suckin face and thinkin it's fuckin dodgy as fuck.
Yes MR hilarious 👦👍👍
great 👌👌👌
totally agree
nice anime figures
hahahah i love this guy
:3
6:41 you could have saved yourself 4 minutes of trouble by just spotting that you have the Laplace transform of sin(2tx)... Anyways, great video
Very nice, but I wish you wouldn't use I in different ways in so many contexts: I, I(t), I(a), ...
How about some subscripts maybe, like this? We want to find the initial integral I_1. Introduce t and define I_2(t) with I_1 = 2*I_2(t=1). Then later define I_3(a) = ... and observe I_2'(t) = I_3(a=0).
good
I challenge you to integrate
(Sin(x²))/x² from 0 to infinity .
Why didn't you just diffrenciete I'(t) another time and you ll get I"(t) and you ll get rid of the x !
Oops it's a year ago and I didn't watch this until today...but uh... I guess that doesn't matter...
Oh no, I lost a minus in a derivative and got a - in front of the answer! I should be more careful next time
Unable to read
name jeff
This time my mind blows up jajaja,
Integrate this pls (sin^2(kx)/x)
Am I alone in finding the notation
sin^2(x)
counter-intuitive? I always think it ought to mean sin(sin(x)).
Is there actually a notation for that?
I suppose you could use f^2(x) where f(x)=sin(x).
sin(sin(x)) is not really a function that shows up "in nature" - of course it's a perfectly valid function in mathematics, but I can't think of a single occurrence of it (or any "nested" trig function) in physics. The reason is that the argument of sine is an angle, and the result is not an angle (more like a ratio of lengths). There just wouldn't be any use for that notation :)
turtlellamacow I've been struggling to find an example and a certain amount of barrel scraping has gone on, but it is possible to nest gamma functions meaningfully so they result in nested sines, albeit not straightforwardly.
Gamma(1-(gamma(0.6)gamma(0.4))gamma(gamma(0.6)gamma(0.4)) =
pi/sin(pi^2/sin(0.6pi))
Approx = -3.85469
I think the symbol that u are looking for it is just "o" I mean
sin(sinx)=fof (x) symbol not f^2 (x) I suggest
wow...
acht
?
it was a joke lol
Neun
🙌
If I cold give a I love it reaction i really would react like that for every single video
The sound is not good as you are speaking in the bathroom.
At 3:55, instead of introducing an exponential function, couldn't you just take the second derivative of i(t) wrt "t" to get rid of the X in the denominator?