integral of sin(x)/x from 0 to inf by Feynman's Technique 

You don't often see a man doing partial derivatives while wearing a partial derivative t-shirt.

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

When you sleep in class 14:01

I see you have finally decided to clothe like a true mathematician, seeing your t-shirt involves partial derivatives. 👌

I really enjoy watching you integrate! Relaxing and fascinating at the same time. Isn't it!

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?

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.

"And once again, pi pops out of nowhere!"

"This is so much fun, isn't it?" Sure.

He's becoming self aware

💀I’m in 11th starting trying to learn this as my physics part needs it.

Believe in Math; Believe in the Pens; Believe in Black and Red Pens.

……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

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!

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.

One of the best math videos I´v ever seen. Changing the function from x to b was a masterpiece.

Who else reads his shirt as "partial asics"?

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.

Now it's time for the Gamma function and some other Euler integrals ;>

I came in just because i saw the name Feynman

K 歌之王?

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!

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!

Using laplace transform and fubini's theorem this integral reduces to a simple trig substitution problem.

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

Thanks for reminding me what I enjoyed about maths! It really is good fun to play around with calculus like this.

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

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!

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?

Please do some putnam integrals They are really tricky and also few tough integrals like these. I love watching your integration videos.

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.

make video on the squeze theorem, I bet you can make it interesting and to show all techniques

the world needs more of this....

Lmao the "isn't it" in the thumbnail

These are so addicting to watch and I don't know why

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.

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.

Beautiful proof, thank you.

I knew this, but it is still awesome. More stuff like this pls!

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?

Best math teacher ever !!!

You said "isn't it" correctly :')

Wow dude.. I thought I've seen it all, and then you FOUND C!! XDDD

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.

00:00-00:23 K歌之王 :D

This comes under differential under integration sign....

just noticed that his shirt is the partial derivative of asics

When he reversed derivative on I(b) by integrating (14:45 min ) and evaluated result as b went to infinity and got zero for that limit-his argument failed. You only get zero if b>0, not if b=0. If b=0 you don't get zero as x goes to infinity-you get divergence

*I S N T I T ?*

There seem to be a few questionable parts about the video. The most important one is at 20:41, it's understandable that as b goes to infinite and x does too, the value is 0, but if x=0 and b is infinity, then you have e^(0*infinity) which is questionable what it equals? Another thing is that how can you assume that b=0 "works the same way" as positive b values, when negative b values don't give a convergent value, and mess up the whole thing? How can you say that 0 which is "on the border" works like positive values. But this probably can be answered easily, but the e^infinity*0 seems to me like no

The cut at 14:02 is kind of confusing.

Can you do the Gaussian integral -integral of e^-(x^2) from -inf to +inf

The kids' laugh made me forget the stress of trying to understanding how you solve it. 😂😂😂😂😂😂😂😂😂

14:30 How can b be >= 0, shouldn't it be just > 0? If it is 0, then the cosine in the solved integral will not converge.

K歌之王wwww 0:00

13:14 fuck that IS so cool

Hi professor, You are doing great...

small question, why cant the integral of -1/1 + b^2 be arccot(b) + c

Feynman was a mathematician, physician and philosopher... super geniuce

Hey Natural Born Improper Integrals Crusher can you please help me integrate [t*e(-sqrt(t)]÷(1+t^2). I used all the methods, except Feynman method and all those online integral calculators, but all of them bumped on it.

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 ?

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.

Love Feynman and this trick was cool and useful. You now have another subscriber :)

14:06 How did that right part magically appear 🥲? Edit 14:54: Ah, post explanation cut 😅…

Please do more video about the Feynman Techniques Thanks a lot

Instead use Fourier transform method, inverse Fourier transform of sampling function is gating function with parameters A and T

Love u darling 😘

I still remember my first year in college. It was filled with so many wonderful moments. This was not one of them.

We can not take derivative into integral when limit is infinite.

But if you allow b to be 0 the integral does not converge... b>=0 does not work, you need b>0.

20:35 Dont you get, if you integrate 0, another constant? Because the derivative of an Constant is 0 too

Holy frick, I never saw that in Calc 3! That's NIFTY! Could you extend this technique to solve the integral sin(x)/x^2 by taking two partial derivatives?

13:19 that is cool

I wouldn't mind more explanations at 10:00... I mean, all the rest is more or less technicalities, but that was the crucial part of the whole thing

Great video. Least I can do is thank you for a great explanation!

Solve it by using Laplace Transform?

Yay i was waiting for this!

why -e^bx * (cosx +bsinx) / (b^2 + 1) = 0? (if x goes to infinity) what if b = 0?

Hey, great video! Loved your explanation. I still have one doubt, however . when we solve for I'(b) and we get an e^-bx in the numerator, the fact that lim(x--->infinity)e^-bx =0 holds only for positive b values, not for b=0. But the issue is, to solve the original integral, we are inputting the value of b as 0, even after taking the above limit. but certainly, the value is matching, so how do we resolve the above anomaly?

Sir wonderful explanation Thank u sir

Red T-shirt? Are you Tom Scott?

Why didn't the intregal change when it's limit to minus

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: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-m0o6pAeCcJs.html "Lobachevsky's integral formula and a nice application." Michael Penn

The "db" in "d/db" looks funny :p

the only problem is that the computation is not justified. Leibniz rule is not stated for improper integrals. If you want it to use for improper integrals, you have to justify all the exchanges of limits that arise.

Wonderful mathematics there sir

谢谢

I'm curious about the very last step. Evaluating atan(0) should yield multiple solutions. How do you reconcile this? What justification do you have for selecting only the principal value?

Isn't I'(b) undefined for b=0? It confuses me how you can make deductions about I(b) at b=0 from its differential when its differential is undefined at that point. Forgive me if this may sound dumb, the furthest I've been taught in school so far is integrating polynomials, but is there a way to justify this in a more rigorous sense or is it actually fine and I'm nit-picking over something irrelevant?

Wow! Great video, amaizing technique. This was an intense integral xD

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 ♡

Me, who is solving frm primitive method and getting the answer 0

My mind was blown infinitely away

I don't get the 13:07 part, if b = 0, x doesn't matter anymore, coz bx is always 0, which makes e^(-bx) = 1, right? And I(0) is exactly what we're trying to solve.

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.

博主听上去会中文❤❤bgm是陈奕迅的K歌之王～

This was great!!!!!

Me watching in 12th class Damn....🙂

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)

I am a jee student can JEE ask that question please tell . I am shocked I cannot think it.