Is it fish or alpha?

@mrakoslav7057

4 жыл бұрын

Use something like Ž than you cant mess up

@ssdd9911

4 жыл бұрын

alfish

@chirayu_jain

4 жыл бұрын

Maybe *alpha fish* 😅

@pwnd785

4 жыл бұрын

Fish of course

@takureido3122

4 жыл бұрын

It is *a* fish

You could also write the cos term as the real part of e^i5x, and then complete the square in the exponential to get the final answer. Physicists use that trick a lot in quantum field theory.

@michalbotor

3 жыл бұрын

f(a) := integral from 0 to oo of exp(-x^2) cos(ax) dx g(a) := integral from 0 to oo of exp(-x^2) sin(ax) dx H(a) := integral from 0 to oo of exp(-x^2) exp(iax) dx H(a) = f(a) + ig(a) ∴ f(a) = Re(H(a)) && g(a) = Im(H(a)) ------------------------------------------------------------------------------------- exp(-x^2) * exp(iax) = exp( -x^2 + iax ) = exp(-( x^2 - iax )) = exp(-( x^2 - 2(ia/2)x + (ia/2)^2 - (ia/2)^2 )) = = exp(-( (x - ia/2)^2 + a^2/4 )) = exp( -(x - ia/2)^2 - a^2/4 ) = exp(-(x - ia/2)^2) exp(-a^2/4) ------------------------------------------------------------------------------------- H(a) = integral from 0 to oo of exp(-(x - ia/2)^2) exp(-a^2/4) dx = = exp(-a^2/4) integral from 0 to oo of exp(-(x - ia/2)^2) dx ------------------------------------------------------------------------------------- i am stuck at this moment. i tried the transformation u := x - ia/2 but i don't know what to do with the integral: integral from -ia/2 to (oo - ia/2) of exp(-u^2) du that has complex limits (i don't know if that is how i was supposed to set the limits of u variable either) and I am not able to split it into two integrals of real variable either. can you give me a hint how can i proceed from here?

@still.sriracha

3 жыл бұрын

@@michalbotor you did all that before understanding the basic concept of substitution :) Exp(-x^2) if multiplied by the euler's theorem would lead to addition of i in the expression whose integral in forward solving is a pain in butt (from past experiences) So moral is to find a logical concept and think on it before just scribbling this is pro tip in competitive level prep. Be well my friend.

@tanmaymishra9576

2 жыл бұрын

Trueee

@groscolisdery1158

2 жыл бұрын

I was going to point it out as my way. But, I guess, the hosts wants to teach the Feynman's method. By the way, Feynman was a physicist if I remember correctly.

@groscolisdery1158

2 жыл бұрын

try y =x+-alpha*x/2

Wow... an integral question solved by partial derivatives, integration by parts, differential equations and the Gaussian Integral to top it all off. Amazing! More Feymann technique questions, please!!

Wow! Feyman’s technique, DI method, Gaussian, ODE all in one. What else can top this? Adding a bit of FTC perhaps

@cpotisch

3 жыл бұрын

It inherently involves FTC because it involves indefinite integrals.

@executorarktanis2323

3 жыл бұрын

What's the full form of ftc?

@BiscuitZombies

3 жыл бұрын

AND the chen lu

@cpotisch

2 жыл бұрын

@@executorarktanis2323 Fundamental Theorem of Calculus. Which there already was plenty of, so I don’t see how OP thinks it was missing.

@executorarktanis2323

2 жыл бұрын

@@cpotisch oh thanks this brings back memories from when I was trying to learn calculus by youtube (self learnt) and didn't know the terms thanks for explaining it now since now I have more broad understanding than what I did 3 months ago

I really enjoy your enthusiasm while explaining things :) Thank you for the videos and please, never lose the energy, liveliness, and passion that you have now. Very nice!

This is the coolest thing I watched today

@thatkindcoder7510

2 жыл бұрын

The coolest thing so far

That's insane!!!!!!!!!!!!!!!!!!!! I love it. It makes me sad they don't teach this in my engineering courses :(

@blackpenredpen

4 жыл бұрын

AugustoDRA : ))) I actually didn’t learn this when I was in school too. Thanks to my viewers who have suggested me this in the past. I haven a video on integral of sin(x)/x and that’s the first time I did Feynman’s technique.

@SimsHacks

Жыл бұрын

It's covered in measure theory (math majors only) as one of the conditions to use the theorem is to find a L¹ function such that |d/da f(x,a)| ≤g(x) for almost all x. L¹ = set of functions with finite Lebesgue integral (not ±∞)

@maalikserebryakov

Жыл бұрын

If you’re sad about that, you don’t belong in engineering. arcane mathematical techniques are nothing but a tool to an engineer, the primary of objective of an engineer is the creative process of ideating new machine designs, and this on its own is a massively difficult issue that takes enormous creative power. If you’re focusing on learning esoteric integration techniques, you aren’t focusing on engineering. I bet you aren’t an engineer now.

@GusTheWolfgang

Жыл бұрын

@@maalikserebryakov hahaha, you hit the nail on the head.

@thesnackbandit

8 ай бұрын

@@GusTheWolfgang Was he right?

I love all your videos, they are hearwarming. Thank you so much !

My initial intuition was to use Feynman to get rid of the exponential term, because if you can get rid of that, trig functions are easy. The thing I didn't think through was the limits of integration: a trig function has no limit at infinity. So quite counterintuitively, it was the cosine that was going to be the troublesome element in all this, while the exponential term was what made the thing solvable.

I've read some of the book's reviews and it looks awesome. I might pick one soon, the applications and integration techniques look interesting

This is one of the most beautiful videos I have seen. ¡Very complete and engaging explanation!

I really wish youtube existed when I was studying mathematics. The potential to be educated in advanced topics without paying a hefty fee for university tuition will hopefully change this world for the better.

A beautiful technique explained beautifully!!

If nothing works to solve a integral Then *feynman technique* would work😉 BTW in the description of book, your name was also there 😁

@blackpenredpen

4 жыл бұрын

Chirayu Jain yup! I gave a review of the book : )))

@roswelcodiep.bernardo7288

2 жыл бұрын

Not that much... Sometimes we need to use complex analysis which includes residue theorem or Cauchy's Theorem

You can also notice that the function is even and replace the integral with half the integral from -inf to inf. Then you break up the cosine into two complex exponentials, separate into two integrals. For each one you can complete the square in the exponent and reduce to the integral of exp(-x^2) by shifting the variable.

@denissmith7671

Жыл бұрын

Niceee 🤤

This was a unique derivation technique. Thank you for sharing.

I found the book in college that Feynman learned this trick from, it's Advanced Calculus By Frederick Shenstone Woods · 1926.

This is great! Thank you! Richard Feynman really was a genius!

Excellent explanation. So brilliantly explained. Thanks a million.

Very nice use of Feynman’s technique. I’m getting the book rn!

@blackpenredpen

4 жыл бұрын

Very nice!! Thanks.

This is such an elegant proof. Really impressive.

It's a bit crazy to call that the Feynmann technique. It goes back to Leibniz and it"s just deriving an integral depending on a parameter. Which by the way demands justification (either uniform convergence or dominated convergence). And in order to make this work you have to be extremely lucky and have a good intuition because you need 1) to find the right parametrization (here it's pretty obvious) ; 2) to be able to integrate the partial derivative for each value of the parameter (which is most of the time not possible) 3) to end up with a differential equation which you can solve (which is most of the time impossible), 4) to be able to compute a special value (here you need to know the value of the Gaussian integral, which is in itself tricky). So, I'd say it's a nice trick when it works but doesn"t qualify as a method...

@JohnSmith-qp4bt

2 жыл бұрын

It looks like the this problem was purposely designed to arrive at an aesthetically pleasing solution. (Given all the justifications/special circumstances/restrictions you mentioned)

@loudfare8840

5 ай бұрын

@@Hmmmmmm487Feynman learnt this method in a random book during his undergrad and he famously showed off to basically everyone that he could solve otherwise very hard integrals.

Maths with you are wounderfull, thanks

I remember this method, because in the video contest I did the integral of (e^-(x^2))*cos(2x) from 0 to infinity. BTW whenever I see e^(-x^2), I always think about feynman technique.

@blackpenredpen

4 жыл бұрын

Chirayu Jain Oh yea you did. And you did a great job on that. : )

@mariomario-ih6mn

4 жыл бұрын

I changed my profile picture recently

@jumpman3773

4 жыл бұрын

@@mariomario-ih6mn Me too

@100kbeforetheyeartwothousa7

4 жыл бұрын

@@jumpman3773 Hi

That was an experience. What a crazy and amazing technique

As a teacher, I loved you saying "negative fish" and will use that in future. Cheers, always good to watch your videos too.

I like it. I love your enthusiasm too.

lovely video, it's this that makes me love calculus

Well I must say ty to you Mr. @blackpenredpen . Thanks to your videos I finished Differential Equations with a B. It was on of my last 2 math classes for my mathematics BS

@blackpenredpen

4 жыл бұрын

Nice! I am very glad to hear! : )

DAYUM, that's one of the most elegant solutions I've ever seen! Why none of my professors was teaching this when I was studying?

Beautiful. Thank you so much.

Beautiful!

Congrats on 400K subscribers!!!

@blackpenredpen

4 жыл бұрын

JZ Animates thank you!!

This is an amazing question for Calc 2.

Because the function is even, you can take the integral from -infinity to infinity and then that would double your answer so the final answer (given alpha = 2) would just be sqrt(pi)/e which i think is even cooler

Nice and clean trick ! Thank you.

I don't have an advanced level of English but that's one of a lot of thing that I love Maths, it's an universal language and your passion in every video is the thing because of I'm still here. Imagine! If I can understand you and I don't speak English fluently, you're MORE THAN AMAZING. Lots of love from Mexicoooo ꒰⑅ᵕ༚ᵕ꒱˖♡

Such an elegant and clever integration technique. Bravo to Feynman and to you, of course. Very cool indeed.

@epicmarschmallow5049

2 жыл бұрын

Bravo to Feynman? For appropriating an integration technique known to Leibniz around 300 years earlier?

@thanasisconstantinou7442

2 жыл бұрын

@@epicmarschmallow5049 ?

One can also observe that f(\alpha) is (up to a constant factor) just the Fourier transform of e^{-x^2}.

That was beautiful man just phenomenal

beautiful, thank you

Very nicely done! Thanks!

Amazing¡¡¡¡ you must record more videos about this topic¡¡¡¡¡¡¡

Awesome vid, really enjoyed it!!!!!!

One of the best crossover episodes ever

Beautiful explanation😀

POV: you can't sleep now, there are monsters nearby 7:36

Love this. Like magic. So beautyful 😱

Great video man!

You didn't got views but all you got is alots of love from the lover of mathematics

this made my day

Absolutely elegant

you can also use the taylor series for cos(5x) and use the gamma function

If there is anything I do not want to forget from my school days is it calculus. Such a beautiful form of math.

Just beautiful!

I love this channel!

@blackpenredpen

Жыл бұрын

Thanks!

Just saw the Gaussian integral=sqrt(pi)/2 half an hour ago in lecture hall. I didn’t know where it came from while my prof was explaining Laplace Transform of t^(-1/2). And now here… What a small world of Mathematics !

Thank you guy.

This was really nice!

I'm halfway through algebra 1, and yet somehow I understand and enjoy most of these videos. You and other channels like you (e.g. Mathologer) make this stuff really accessible, and importantly, fun. (Not to say I don't enjoy my algebra 1 class!)

@skyrider8890

4 жыл бұрын

A great teacher is everything, right?

Sometimes, a lot of integral practices makes me to say Instagram as Integram

@blackpenredpen

4 жыл бұрын

lol!

@NXT_LVL_DVL

Ай бұрын

Instagram is the culprit

A fun trick would also be using the fourier tramsform of the bell curve

π and e in a same expression is always beautiful

convert cosine to sum of exponentials, complete square, Gaussian integral is root pi, can do it in your head in a few minutes, under a minute if you're confident, and almost instantly if you've seen a few of these. Cool to see feynman's technique at work though, great video!

👀great work sir

When you set alpha equal to sqrt(2 - 4ln(2)), you get sqrt(pi / e) for the answer. Pure beauty indeed.

All the feynman's techniques are UNIQUE 👍👍

such a beautiful video thanks

11:20 ...............the sentence is veryyy TRUE indeed !!!!

I very much enjoy watching the derivations, even though I know I'd probably never be able to figure it out myself.

Wow, nice solution !!!

University teacher: ur exam has integrals The intégral during the exam:

Thanks for remind me of a feeling in Maths again. It is 20+ years ago since my college course. Actually, I love Maths, in the past It is my recreation. But living in real life, I have no time to solve the challenge Maths problem. Life is hard.

I am agreeing that Feynman's technique is having a good strong hold in solving exponential integrals...but rather than complicating we could have solved it by manipulating "cos(5x)" as (e^5ix + e^-5ix)..it also saves the time...

This is such a pog method and this vid is amazing

I wish I understood. Someday, maybe. Man that's orders of magnitude beyond what I can comprehend at the moment.

I notice the *Feynman' technique* (aka. _Leibniz Integral Rule_ ) depends basically upon parameterizing the parts expansion here; its the _by-parts_ part that gives it the power in my opinion for what its worth!

Thank u 💞

This is amazing

Fabulous!

That was a wild ride!

Nice don't remember running across the Feynman technique before.

you can also use a Fourier transform

Graet integral! Feynman is a genius

int lnx/(1+e^x) from 0 to infinity

Just wonderful 🤩

Amazing!

This is the hardest integral I’ve gotten right on my own! So proud of myself

@shivanshbellamkonda6898

3 жыл бұрын

Integrate (e^x)(x^x)(2+logx) wrt to x Please someone do this

@ericwilliams1832

Жыл бұрын

x^x oh no

Great "sir"

Very Nice.Thanks.

Can't say I understand, but I do agree: it's very nice!

Feymann's Technique + Differential Equation

@blackpenredpen

4 жыл бұрын

Mokou Fujiwara yes. And Chen lu!

11:38 I love how satisfied he looked after all that he did.

This is beautiful! I love Feynman's method!

Even more beautiful: Since e^(-x^2) cos(2x) is an even function, the integral from -inf to inf just becomes sqrt(pi)/e.

the sinx over exp x^2 when x goes to infinity needs a bit more rigor when calculating, you can't just say it's a finite number on the nominator (max +1 or min -1) because the lim of the sin function when x goes to infinity doesn't exist. I believe one way to alleviate this, is by using the "sandwitch" theorem; wikipedia -> Squeeze_theorem

@elliotlacinai3041

2 жыл бұрын

Sure but I think it's safe to assume that if the viewer understands Feynman integration, they also know (or intuitively understand, at the very least) why the expression evaluates to zero at inf

very good proof amazing use of differential equations

@jacobbills5002

3 жыл бұрын

Just watch this impressive Math channel kzread.info/dron/ZDkxpcvd-T1uR65Feuj5Yg.html

Very satisfying

Very nice!