integral of sin(x)/x from 0 to inf by Feynman's Technique
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- Опубликовано: 3 окт 2024
- The integral of sin(x)/x from 0 to inf by using Feynman's technique (aka differentiation under the integral sign). This integral is also called the Dirichlet integral. Check out another example of Feynman's technique of integration: • Feynman's Technique of...
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integral of sin(x)*e^(-bx), • The appetizer, integra...
Another example, Integral of ln(x^2+1)/(x+1) from 0 to 1 by Mu Prime Math, • It took me 3 hours to ...
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This is so famous, i still remember 8 years ago, when my uni professor told me, there is psychiatric hospital for those who still try to find a primitive of sin(x) / x... lol
Kkkkk
Lol
For some reason , “lol” looks like mod(0)
@@camkiranratna do you mean absolute value?
@@deltaspace0 Absolute value is also called mod in some places.
I really enjoy watching you integrate! Relaxing and fascinating at the same time.
Isn't it!
PompeyDB it is!
it is, is not?
It's!
I really enjoy watching you disintegrate! Relaxing and fascinating at the same time.
Isn't it!
@@rehmmyteon5016 lmao
I recall doing this integral many years ago. Back then we used contour integration. We chose the contour to be a semi-circle of radius R centered at the origin . The origin was indented and cotoured with a semi-circle of radius r. The semi-circle was located in the upper-half of the Cartesian plane. Complex integration in one of the most potent methods for dealing with such problems.
I agree, I solved this too in my first course of Applied Mathematics in college where we used complex analysis techniques
ruclips.net/video/Ff4LRlflib0/видео.html
Yeah that's true, that's how I learnt it/saw it first
Integrales cerradas en variable compleja?
You can do integrals on complex bounds (lower/upper) 😮? Or is it Real bounds but integrated on Complex functions?
@@louisrobitaille5810 complex functions and complex bounds. Turns out that the path you take *mostly* doesn't matter!
You don't often see a man doing partial derivatives while wearing a partial derivative t-shirt.
hahahahaha! honestly, that wasn't planned.
blackpenredpen I just realised after reading this...
me 2 lollll
It seemed to me like some sort of band sign like Nike at first
what’s the difference between partial derivative and normal derivative?
When you sleep in class 14:01
More like when you blink in class :)
but the answer was spoiled in that part :D
when you struggle not to sleep
Idk why was the video cut? lol
Ahnaf Abdullah I wanted to add that explanation why b has to be nonnegative
One of the best math videos I´v ever seen. Changing the function from x to b was a masterpiece.
Yes, Feynman was a brilliant mind
The part where the constant C is determined by checking the limit of the function at infinity is very elegant. Beautiful proof. Of course, there are a lot of technical details that mathematicians would think about (is it correct to derivate inside the integral, exchange limit and integral, etc.). But this video is a great summary of the overall strategy. Very nice work!
.l
I see you have finally decided to clothe like a true mathematician, seeing your t-shirt involves partial derivatives. 👌
MeowGrump lolllll this is a good one!!!
asics = "Anime sane in corpore sano,"
"Sound mind/spirit in a sound body."
Anima not anime (but that's somehow relevant :))))
👌 looks like the partial derivative sign XD
So, it is soul eater then?
Hi, I just learned this technique over the summer. I was amazed. I used it to solve a problem from American Mathematical Monthly. It was fun, not only sending in a solution, but learning this amazing technique used by Feynman!
Wow. At the begining the integral with the exponential function looks more complicated, but that function allows to have a closed form and the Leibniz theorem is fundamental. Great work!
Lies again? So fat
you told us not to trust wolfram and now you confirm your answer in wolfram. what am i supposed to do with my life now?
Dokuta Viktor trust no one
Dokuta Viktor Ask wolfram.
Dokuta Viktor only if it gives the same answer as what we got.
blackpenredpen what if what you got is by looking at Wolfram????
then don't get things from Wolfram but just check your answer with it.
"And once again, pi pops out of nowhere!"
him: "And now let's draw the continuation arrow with also looks like the integration symbol. That's so cool."
Me: "Ha."
I happen to remember just enough calculus to follow along. Interesting. Thank you.
Sean Clough yay! I am happy to hear!
i am 17 years old and i am from india .............i am able to understand it clearly ......thank you sir , love you and your love for mathematics 😊
Feynman was a mathematician, physician and philosopher... super geniuce
Physicist*
@@juanpiedrahita-garcia5138 lol
genius , even as a joke it hurts my eyes
the world needs more of this....
You always manage to make me click to watch you do integrals I've already done long ago!, but this integral of sinc(x) was really gorgeous. It's kinda the method for obtaining the the moments of x with the gaußian. I hope to see more of this kind.
Thanks for reminding me what I enjoyed about maths! It really is good fun to play around with calculus like this.
Wow, that was an heavy load! I never saw anything like that before...it'll take me a few days to digest the technique. Well done!
That's what she said
This problem can be simply solved using complex integral (getting the answer directly without a piece of paper). However, I’ve to admit that the method introduced here is VERY SMART. Thank you!
Best math teacher ever !!!
How the heck do 2 people that didn't know eachother ' invent' calculus at the same time.Simply fascinating. This was awesome to watch, I now have a better understanding of how partial derivatives work. I now must go back and study calc shui I can come back and fully digest this.
18:12
I like the idea that, after going through all that, we figure out that the integral from 0 to infinity of sin(x)/x dx is equal to...
Some unknown value.
its not that unexpected though if you look at the function... its just looks very convergent.. (this can ofc be very deceiving)
@@antonquirgst2812 But there's the fact that as x grows larger, it tends to 0 because sin's at most 1 or -1.
@@createyourownfuture5410 yup - totally agree - x grows linear while sin(x) is periodic!
@@antonquirgst2812 Aaaand it approaches 0 from both sides
@@createyourownfuture5410 It actually approaches 1 from 0
I can't believe I just spent 20 minutes watching a video about integration and loving every second of it. A few years ago, I used to despise Maths
Great video using Feynman's technique but would never tackle this integral in this way. Once you've applied the Laplace transform it's much easier to use Euler's formula and substitute sin(x) with Im (e^ix). Haven't read all of the comments but I'm sure this has already been mentioned
I'm familiar with using the Fourier transform to find the integral, but I don't quite see how you'd use the Laplace transform.
@@Sugarman96 - the Laplace transform is what the above video uses when creating his function I (b)
@@Sugarman96
I'(b) is the same negative laplace transform of sin(x) which you can use to easily find I'(b) instead of doing whatever he did.
Using laplace transform and fubini's theorem this integral reduces to a simple trig substitution problem.
"This is so much fun, isn't it?"
Sure.
lol
I started to get interested in mathematics after seeing this integral before!
Thank you for giving me the solution :)
He's becoming self aware
How so? What did you notice?
isn't it?
His enthusiasm is contagious wish he was my calc professor back in the day I would have loved that class
Please do some putnam integrals
They are really tricky and also few tough integrals like these.
I love watching your integration videos.
I found another way to solve his problem that feels more unique, alhough your solutions is much more straightfoward and intuative.
I started by doing everything the same up until you get to I'(t) = -integral of sintheta times e^(-t*theta)d theta. Afterward, I turned sintheta into Im(e^(i*theta)). Hrn I used exponent laws to combine the exponentials and and take the integral from 0 to inf. Then I took i tegral on both sides and evaluated I(inf) to get c=0. Then I evaluted I(0) = -Im(ln(0-i)) = pi/2.
💀I’m in 11th starting trying to learn this as my physics part needs it.
Just found your video from randomly browsing youtube, and I really like your enthusiastic way to explain those problem.
I heard about this differentiation technique since I was a sophomore, but didn't get the "why" part: Why differentiation? Why new parameter? Why e^-bx? It's all make sense to me now thanks to your video. Keep up the good work!
Thanks Donny. You can also check out Zach's page in my description. He has a lot of great stuff there!
The best ever demonstration i've seen.
I always thought this integral to be done by an algorithm based on the sum of trapezium areas which gives approximatively the same result as pi/2.
Really amazing demo.
The next question would be what is the primary function of integral of
sin(x)/x dx ?
Fantastic video! I was thinking literally just the other day that I hope you'd make a Feynman technique video and, as through magic, here it is! Would really love to see more videos about alternative / advanced techniques.
Believe in Math; Believe in the Pens; Believe in Black and Red Pens.
yay!!!!
Yes, i did It and i got 10 in my integral calculus exam :') two months ago !
@@MrAssassins117 now 3 years ago lol
@@pranav2119 now 3 yrs and 14 hrs ago.
Instead use Fourier transform method, inverse Fourier transform of sampling function is gating function with parameters A and T
Now it's time for the Gamma function and some other Euler integrals ;>
You’re awesome bro, thank you for such a clear video. And leaving a link to where you first saw the method is very classy, I respect that. Greetings from the US, my friend 🙌🏽🎊
Great videos, planning to recommend to my students but not a fan of notation x=inf or of plugging in x=inf. Students will do this without the understanding you have and will lead to some issues in calculating limits such as inf/inf =1. Please remember you're a role model :)
Deanna Baxter I always just plugged in infinity. Didn't lead to any misunderstandings. It's more cumbersome to take the limit, though it's technically correct. You first introduce indeterminate forms in order to avoid issues.
Rudboy
I agree, sadly sometimes students won't be lucky enough to get a grader who will be forgiving.
I one time did that and the grader goes
"While your final answer is correct, you can't just set something as infinity"
There was another part of the problem where I got the answer correct, and they go "your answer in this part is correct *AND* your math is right, but you weren't supposed to get it that way"
I ended up getting only half credit for that problem
This was an assignment where we had to do ten problems but only *two* of them would be selected at random and graded so one quarter of my grade on that went out the window
Needless to say, I was salty
Deanna Baxter if the students are interested in this integral in the first place, they should be ok and understanding this shorthand notation. Btw, a MIT professor also does that in his calc lectures for improper integral.
Here ruclips.net/video/KhwQKE_tld0/видео.html
Thanks for the comment and thanks for watching!! :)
I still remember my first year in college. It was filled with so many wonderful moments. This was not one of them.
I knew this, but it is still awesome. More stuff like this pls!
Great video. I've seen many of yours. You're doing a great job speaking about unusual techniques and methods in Calculus.
make video on the squeze theorem, I bet you can make it interesting and to show all techniques
Paul Montes dr. Peyam is actually going to do that soon
One of the best videos on this great channel. Beautiful.
At 14:18 : You say that since e^-bx matters, the integral converges for all values of b >= 0. Well it's true for b > 0. The reasoning cannot work for b = 0 because it's slightly more complicated than that (but it converges too).
Counter example : Integral from 0 to infinity of e^-bx/x dx doesn't converge for b = 0.
Hi, this is Zachary Lee.
You are absolutely right to be concerned about the convergence at b=0. What you want to do is let b approach 0 from the right. If you want a rigorous explanation, check out Appendix A, on page 21 of this document:
www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf
Footskills here's the man!!!
Yeah that was a brief explanation haahahhahaha
And here I am again!!! Btw, great explanation!
PackSciences your counter example should be rearranged as (e^(-b x)-1)/x
Btw e^(-b x)/x diverges for all values of "b"
You rock man! These are a great set of videos for young aspiring mathematicians!
Love Feynman and this trick was cool and useful.
You now have another subscriber :)
Wonderful!!!!!!!!
I love this video, for many reasons.
When I watching it, I just enjoyed.
Thank you so much for this.
Great video. Least I can do is thank you for a great explanation!
Thank you!!!
It's the first time I see this way of integration and I'm amazed!
Does theta stand for anything particular in Greek, relating to angles? Or is it just an arbitrary letter that has historically been used for representing angles similar to how x and y represent Cartesian coordinate variables?
Probably, the reason x/y/z are used for representing Cartesian coordinate variables, is that it is the trio of neighboring letters in the alphabet, that is LEAST likely to stand for anything in particular, and therefore they are letters used as wildcards.
The content on your page is always so informative, and your excitement for the math you show is contagious. By the way, have you considered making a Patreon page? I would gladly support!
Also, how sneaky of you to wear the "Basic" shirt that has the lowercase-delta on it, foreshadowing the partial derivatives you use in the video.
LOL! Thanks!
In fact, that wasn't planned. lolllll
Thats not a lowercase Delta
Here comes the paid publishing
Can you like a video twice? Just watched this again, and still awesome. Thanks for this!
Great video. The e^(-bx) looks random until you realize that lots of these problems use the same parameterization.
The answer is actually 42 though. Proof: summing the positive and negative regions under the curve we get a conditionally convergent series. Add positive terms until you exceed 42, then add negative terms until you go below 42, then add more positive terms until you exceed 42 again, etc. The sum will converge to 42 so this is the value of the integral. QED.
sin(x)/x? More like "Super derivations that are always the best!" I know a lot of other comments say it, but I think this technique is just so cool, and it can take things beyond a lot of other integration videos. Thanks for sharing!
Beautiful proof, thank you.
bruno edwards Yup, leibniz rule is very powerful.
Thanks my man I've been trying to solve that question for a long time byparts and some other methods didn't get it thank you I'm a big fan 😍
Thank you for showing the trick with the e-function. Would not have seen this and could be very useful. When I did this problem for -inf to inf I did it with Fourier transformation by writing sinx/x as the fourier transformation of the rectangle function. After changing order of integration you get a delta distribution and the other integral collapses as well. Of course you get Pi at the end.
To be fair, Zach showed me (as I mentioned in the video).
We used to think that it is such a basic calculus skill for all college students, now it becomes a show and privilege. I hope it will bring more interests among the young generations.
Can you recommend a good proof of Liebniz Rule to follow?
It seems like one of those simple/obvious things that would actually have an interesting/ instructive proof.
All the computations are only valid for b>0, because you need the exponencial to derive inside the integral under Lebesgue's domination Theorem. But at the end you do b=0. One further step is needed to show that I is continuous at 0. Note that this os not easy because |sin(x)/x| is not integrable, and therefore you cannot use standard continuity theorems as they require a domination hypothesis.
Hello Alejo. Yes, I agree, but that is exacly my point. You need a more demanding theory (such as Denjoy integrability, among other possibilities) to justify the calculus presented in the video.
Correct me if I'm wrong, I'm a bit rusty, but don't you need to prove uniform convergence before bringing the differentiation sign inside the integral?
I think you're right. I was thinking the same
Does it help? I am not an expert in the field (yet):
en.wikipedia.org/wiki/Leibniz_integral_rule
I love your enthusiasm and your clear explanations. Thanks!
The kids' laugh made me forget the stress of trying to understanding how you solve it. 😂😂😂😂😂😂😂😂😂
Mazed Suhaimi yay!!!!
Amazing, such an elegant way to solve that integral, I'm a physicist and it helped a lot! Thank you!
These are so addicting to watch and I don't know why
Important function, also good for Interpretation of some Integrals with Delta Distribution. So it has a practical use as well.
Great Video, thanks and Keep on with this interesting and usefull stuff, …. makes lifes a lot easier at work.
Hi professor,
You are doing great...
Thank you!
Excelente apresentacao ! Sempre usei esta tecnica sem saber q se chamava de tecnica de feynman ! Vivendo e aprendendo !
Lmao the "isn't it" in the thumbnail
That was the most peaceful boss music I've ever heard. And it's definitely boss music when you're trying to integrate sin(x)/x
Please do more video about the Feynman Techniques
Thanks a lot
Jack Chai ok
more integrals using differentiation under the integral sign please!
For the ones that want to dive into the details, I think we have to justify that the differential equation is defined for b in (R+*) in order for e^(-bx) to actually tend towards 0, then use the continuity of parameter integrals so that I(b) -> I(0) when b->0. Finally, the dominated convergence theorem gives us that I(b) -> 0 when b->inf. We conclude with the fact that arctan + pi/2 -> pi/2 when b->0, and uniqueness of the limit : both limits I(0) and pi/2 are equal ♡
Why would anyone think to add e^x thiugh this COMES OUT OF NOWHERE..what I thought to do was replace sinex with e^ix from Eulers formula..isn't thst smarter and more intuitive? I think he needs to justify where e^x cones from if anything it should be ln x he is adding nkt e^× since 1/× is the derivative of ln x not e^×..
Thanks SIr.. U explain things in a great manner that even i could understand, thanks for solving the qsn stepwise
The cut at 14:02 is kind of confusing.
thomas g Just got to it, I reckon he's solved it already, then started talking about his steps and realised it'd fit better with the part where he was previously (in his timeline) talking about it.
It's so he can outline that b has to be positive, and it probably makes the most sense to put the clip here
When u sleep on class
it spoiled the rest of the video
"so lets draw the continuation arrow, which looks like an integral sign, that is so cool"... friends, this guy is pure gold !!!!
My mind was blown infinitely away
伊藤那由多 loll
Great video!great technique! Great explanation! A huge hug from Peru - South America
Yay i was waiting for this!
This is the Dirichlet function and the Feynman technique is great way to solve it. Downside of Feynman technique is you cant plug and chug. The formulas have to be checked along the way for validity . Such is life. Thank you Pen(Black + Red)
20:35
Dont you get, if you integrate 0, another constant? Because the derivative of an Constant is 0 too
elp 486
It’s a definite integral of 0 from a to b, so there’s no area. : )
There's a simpler way of calculating this integral. This funcion is really famous, is the sinc function, and is the fourier representation of an ideal low-pass filter, a rectangular function.
The integration property of the Fourier transform tell us that the integral from minus infinity to infinity of a function in the time domain is equal to the frequency domain (or Fourier domain) representation of the function evaluated in 0.
So, to calculate this integral, you just calculate the Fourier transform and just evaluate in 0, which gives you Pi. Of course, because of the integration limits, you get the result divided by 2.
this result is used to prove the convergent of fourier series though
I came in just because i saw the name Feynman
me too
You got it ,me too
You said b is greater than or equal to zero. But if b is equal to zero, then the limit when x goes to infinity e^(-bx) will become 1 and in cos(infinity) there no limit.
What's also very Interesting, we could also use *Lobachevsky's integral formula* :
*integral from 0 to +∞ of [ f(x) * (sin(x) / x) ] = integral from 0 to (π/2) of [ f(x) ]*
So our example:
integral from 0 to +∞ of [ (sin(x) / x) ]
has *f(x)=1* :)
Now we use Lobachevsky's integral formula:
*integral from 0 to +∞ of [ f(x) * (sin(x) / x) ] = integral from 0 to (π/2) of [ f(x) ]*
integral from 0 to +∞ of [ 1 * (sin(x) / x) ] = integral from 0 to (π/2) of [ 1 ]
integral from 0 to +∞ of [ (sin(x) / x) ] = integral from 0 to (π/2) of [ 1 ] = x | computed from 0 to (π/2) = (π/2) - 0 = (π/2)
*Answer:*
integral from 0 to +∞ of [ (sin(x) / x) ] = *(π/2)*
Mr Michael Penn made a video (entitled )
where he calculates that example using Lobachevsky's integral formula:
ruclips.net/video/m0o6pAeCcJs/видео.html
"Lobachevsky's integral formula and a nice application."
Michael Penn
You really save life via RUclips
Your claim that the expression inside the integral is going to 0 when x approcheing to infinity is very problematic when you understand that we considering the case when b=0. Then, the integral wouldn't be convergent, so how can you explain that?
x approaches 0 from the right. With a weapon. Also discussed later in the comments
Keep going my friend... the method that you're using to explain things is great
Who else reads his shirt as "partial asics"?
Easier than solving for C is to write I integral from 0 to b of I' = I(b) - I(0)
Left-hand side is ok even if you use a different antiderivative, as long as the choice on the left is self-consistent. Then you can take limit b to infty and solve for I(0)
This integral's easy. Just pretend that all angles are small, replace sin(x) = x, the x's cancel so you're left with the integral of 1 :D
Hard to justify with those zero to infy limits. ;-)
so, pi/2 \approx inf?
How can you pretend all angles are small? The angle goes to infinity o_O
@kikones34 : Yeah, that's the joke (note the ":D" grin at the end.). But it _does_ work for the _variable_-bound integral
int_{0...x} sin(t)/t dt
which, by the way, defines the standard mathematical function Si(x), the "sine integral" function, because you can then consider when all angles in the integration are small. If you take sin(t) ~ t then you say for _small_ x that
int_{0...x} sin(t)/t dt ~ int_{0...x} t/t dt = int_{0...x} dt = x
so Si(x) ~ x when x is small. And a Taylor expansion will show you that that makes sense, too:
Si(x) = x - x^3/(3.3!) + x^5/(5.5!) - x^7/(7.7!) + x^9/(9.9!) - x^11/(11.11!) + ...
so the first (lowest-order) term is x, thus at small x, Si(x) = x + O(x^3), meaning the rest vanishes like x^3.
@mike4ty4 Oh, sorry, I totally didn't get you were joking. I've been on a RUclips trip of flat earther videos before watching this, so I was in a mindset in which I assumed nonsensical statements are actually serious and not jokes xD.. D:
You are better than my professors!
is there an integration bee where you teach? i think you'd be the guy to create a lot of fun (and probably cruel) integrals for students x)
Based on the fact that english doesn't seem to be your native linguage, I think you should speak more slowly, pronouncing carefully and completely every word. Sometimes it is VERY dufficult to understand what you say. Apart from that, your videos are excellent; you have a great mathematical knowledge and you are a great teacher. Well done!
……人又相信 一世一生這膚淺對白
來吧送給你 要幾百萬人流淚過的歌
如從未聽過 誓言如幸福摩天輪
才令我因你 要呼天叫地愛愛愛愛那麼多……
If you know you'll know
Of course I know 😆
ahhh, that's why the intro song is so familiar, k歌之王 by Eason Chan!
My favorite mathtuber
You said "isn't it" correctly :')
I did, isn't it!
hehehe
@@blackpenredpen didn't I
@@SanjeevKumar-js4mu lol
good old sinc function. Learned about it last year in signals and systems. Always nice to have refreshers like these that explain everything so well. Good job!
The integral over sinc(x) also has a name (since it's not an elementary function), the Si(x) ("integral sine", no the abbreviation doesn't make sense).
What does the c stand for in why it is called a sinc function?
@@carultch it’s just a notation that is used to define sin(x)/x. I forget if there are any special properties to it, but it was used a lot in the signals class I took years ago during my undergrad. I believe I watched this the semester after I took it. Taking a look a bit, pretty much the application we had was that the Fourier transform of a rectangular function is the sinc function. I don’t remember much past that as I haven’t used it for years since then
@@UnOrdelyConduct I found the answer. It is called "sine cardinal". Not sure what cardinal would mean in this context, or if it has anything to do with cardinal numbers, but that's why it is called sinc of all possible names.