Using Feynman's technique TWICE! (the integral of sin^3(x)/x^3 from 0 to inf)

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Using Feynman's technique TWICE! (the integral of sin^3(x)/x^3 from 0 to inf)

Рет қаралды 246,435

Жыл бұрын

We will evaluate the improper integral of sin^3(x)/x^3 from 0 to infinity by using Feynman's technique of integration (aka differentiation under the integral sign, Feynman's integration trick, or Leibniz's Rule). This is definitely one of the coolest integration techniques (but unfortunately it is not often taught in a calculus class).
See the integral of sin(x)/x from 0 to infinity: • integral of sin(x)/x f...
integral of sin^2(x)/x^2 from 0 to infinity: • 100 integrals (almost ...
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Пікірлер: 207

@blackpenredpen Жыл бұрын

This is from the 100 integrals part 2. See the full video here kzfaq.info/get/bejne/oLeqZMqCl5qzeJc.html

@lorenzosaudito Жыл бұрын

I was wondering why you looked so tired, now I understand 😂

@wonghonkongjames4495 Жыл бұрын

YES MU PRIME THINKS OUTSIDE THE BOX ALSO DO THE MERITAVIEN AND MIND YOUR DECISION TOO THEY ARE OUTSTANDINGLY DIFFERENT AND DR PEYAM TOO

@RaKeShCHauHAN28021 Жыл бұрын

I Watch your video from India

@lordofhunger5175 Жыл бұрын

We need a tutorial about where to use each pen

@donkeybros8734 Жыл бұрын

He has a KZfaq short

@geneyoungdho Жыл бұрын

Characteristic part

@patrickinocencio2305 Жыл бұрын

agree

@mathssucker Жыл бұрын

hahaha I have a tutorial!

@Mehmetbilendegilypo Жыл бұрын

youtube.com/@mehmetturk6583

@weinihao3632 Жыл бұрын

This kind of video, where you show your thought process and consider which route to go and even hit a dead end is very very nice as it teaches how to tackle the problem instead of simply presenting a deus ex machina solution.

@julianw1010 Жыл бұрын

I really like the deus ex machina solution too though

@maalikserebryakov Жыл бұрын

@@julianw1010 who asked?

@CurryMuncher2 7 ай бұрын

@@maalikserebryakovhuh?

@LuigiElettrico Жыл бұрын

Looking at the clock and hearing it being synchronized with my own wall clock makes me feel like I am in the class :D Great integral!

@vatsalmalav440 Жыл бұрын

I like how you say "this guy" making numbers look like living things that make your life easier and many times Hard. This is a great integral you solved I loved it.

@yutaj5296 Жыл бұрын

Expressing sin³(𝑥) in terms of sin(3𝑥) and sin(𝑥) using the triple angle formula in the first place seems helps.

@zunaidparker Жыл бұрын

If you Laplace transform this integral you'll see why the value for the 3rd power is different from the first two powers. Essentially you're doing a convolution, which amounts to taking a moving average over a sliding window of a rectangular function. For the first 2 powers, the window isn't wide enough to affect the value of the average over the moving window, but for the 3rd power, eventually we are averaging zero contributions from outside the rectangle which brings the moving average down. 3blue1brown did an AWESOME video into this: m.kzfaq.info/get/bejne/bpthiJhmnNCaeHk.html

@h10r60v 3 ай бұрын

14:30 man i know that happiness and you have to experience it atleast once in a lifetime!

@jamiewalker329 Жыл бұрын

The integral - after using the fact that the integrand is even, using the triple angle formula can be written as 1/8 Im{ integral (e^i3x - 3e^ix)/x^3 dx } where the integral runs from -infinty to infinity. We can analytically continue into the complex plane, separate the two integrals, run a contour along an infinite semi-circle in upper half plane, and a small-semi circle in upper half plate, circulating the singlarities at z = 0 of the function. Using Jordan's lemma to determine that the integral around the large semi-circle is 0, and using Cauchy residue (no poles within contour) means that the integral is equivalent to integrating in the complex plane the above integral around an infinitely small semi-circle, centred at z = 0. The result is 1/8 Im(0.5*2*i*pi*residue at z = 0). The residue, of the above integrand at z = 0 is -3 (which can be quickly checked by expanding the exponential numerator to quadratic term. Plugging this in gives the answer...no Feynman...

@pashaw8380 Жыл бұрын

Indeed.

@chayanaggarwal3431 Жыл бұрын

Yes I did thought of the same way whenever the limits are till infinity with some power of x in denominator I always first try to use the residue theorem

@peamutbubber Жыл бұрын

Except u can do this without any of that, u overcomplicate the simple

@CliffSedge-nu5fv Ай бұрын

Well, yes, _obviously_ any Calc 1 student already knows how to do that, so why not challenge yourself to trying a different method?

@drpeyam Жыл бұрын

Reminds me of the Borwein integrals a bit

@sngash Жыл бұрын

This is excellent and the video led me to Math 505's generalized version of sin^n(x)/x^n which looks like a great beast for you to work your magic on and possibly make understandable at around calc 2 level :). I struggled following the differentiation portion

@fordtimelord8673 Жыл бұрын

I know it’s not the same integral, but I recommend checking out the use of complex contour integration to come up with a general formula for the integral of sin (x^n)/x^n on the same interval. kzfaq.info/get/bejne/pdyaapSi1Z2nZJ8.html

@omograbi Жыл бұрын

10:14 shouldn't it be sin(3tx)/x? Or it's anyway the same answer?

@ianfowler9340 Жыл бұрын

I would say it's a different answer.

@MarkPaul1316 Жыл бұрын

@omograbi gives the same answer, but he could have continued with sin(3tx)/x, making the substitution a = 3tx, arriving at the integral of from 0 to infinity of (sina)/a which gives pi/2.

@yoyoezzijr Жыл бұрын

its the same answer, integral of sin(tx) / x from 0 to ∞ is π/2, so putting 3t instead of t will be the same

@wolliwolfsen291 Жыл бұрын

Yes, it‘s the same result, but it is confusing

@krisbrandenberger544 Жыл бұрын

Both integrals will have the same exact value. Performing the substitution u=t*x for the first one will imply that 1/x=t/u and dx=(1/t)du, which makes the t's cancel out. Likewise, for the second one, if you let w=3*t*x, that will imply that 1/x=3*t/w and dx=(1/(3*t))dw, which makes the 3*t's cancel out.

@ritvikg Жыл бұрын

10:16 I didn't get this step. Firstly how did he replaced the sin(3tx) with just sin(tx) and in the next step after substituting tx as u, he should get a 't' after integration which he missed as well. It should have been -3πt/8 + 9πt/8 considering his previous step of omitting 3 from the sin is right. Can anyone help me with this.

@ernestschoenmakers8181 Жыл бұрын

I didn't get that either, he switched from sin(3tx) to sin(tx), must be a mistake over there and maybe despite of that the result is the same by coincidence.

@digbycrankshaft7572 Жыл бұрын

The first issue is definitely a mistake. With regard to the u=tx substitution this gives dx=du/t. When the substitution is performed it gives integral of sinu/(tx) du and as u=tx this gives integral of sinu/u du with the limits of integration being u=t×0=0 and u=t×infinity=infinity. This then is just the straightforward known integral with u instead of x with the same limits of integration giving pi/2.

@amirbasson532 Жыл бұрын

There was no mistake, The integral from the type: integral from zero to infinity of sin(Ax)/x dx always equal to π/2, because: (Integral from zero to infinity of sin(tx)/x dx) let tx = u x = u/t dx = du/t and then: the integral of sin(u)/(u/t) × du/t equal to: t×sin(u)/u × du/t t and t cancel out (the integral from zero to infinity of sin(u)/u du)= π/2 and because of this: the integral from zero to infinity of sin(3tx)/x = the integral from zero to infinity of sin(tx)/x = π/2 Hope I helped you :)

@digbycrankshaft7572 Жыл бұрын

@@amirbasson532 it was a mistake not making any reference to this fact as it was an assumption which has evidently caused confusion to several people.

@shoto206 Жыл бұрын

@@amirbasson532 ohhhh that explains it, thanks!

@-fai7485 Жыл бұрын

Hey sir, Feynman's technique is mad cool but... Where should I set the parameter? Is there any "rule" to follow? I mean, you are supposed to put the parameter on a place in which after deriving, the integral is easier to solve, but it would be marvelous if you have a structured guide that tells you where to put it depending on the situation. It would be great a video like... "When and where to use Feynman's technique" Thanks sir.

@abdulmalek1118 Жыл бұрын

Hello ! I hope you see my comment I saw this nice question so that I recommend it The question is : solve the system of equations a = exp (a) . cos (b) b = exp (a) . sin (b) It can be nicely solved by using Lambert W function after letting z = a + ib Hope you the best ... your loyal fan from Syria

@vogelvogeltje Жыл бұрын

You have 60hz hum coming from your microphone JSYK. Try to turn your microphone up more without clipping over 0dbFS, or look and see if any wires are crossing over a power wire from your interface.

@pacifyplayer 4 ай бұрын

After you simplified I''(t), where you split the integral into two integrals, how did you get rid of the 3tx inside of the sin function? In the next step, there is just a tx and no 3tx, how can you do this? Can someone explain, please?

@mumilala9940 Жыл бұрын

The alternative way is Fourier transform, split it into (sinx/x)(sin²x/x²) then convolution time!

@md2perpe Жыл бұрын

I used that technique for the integral of sin²x/(x²(1+x²)), seen in kzfaq.info/get/bejne/iZtid8Sh1K6VZ4E.html

@CDChester Жыл бұрын

Another integral for the collection!

@user-rw5eh8mi9f Жыл бұрын

Because sin^3(x)/x^3 is an even func you can write the integral to be 1/2 of the same integral over the real line. Then after you take the d/dt and use the sin^2(x)=1-cos^2(x) you get an odd function over a simatric interval, so it's 0. So you don't need to take the second d/dt you did

@wcottee Жыл бұрын

Maybe I missed it but at 10:16 how did the second term go from sin(3tx) to just sin(tx)?

@ActALCOCERBONILLAARTUROAZAEL Жыл бұрын

Same doubt

@noopcode Жыл бұрын

it was a mistake but the integral is still pi/2

@cristofer6806 Жыл бұрын

yeah It should be 3tx but as he already mentioned, the result is π/2 regardless of the input.

@selectname9790 Жыл бұрын

I don't think we need to do the u-substitution separately for the sin(3tx) to see that it is also π/2. We can reason directly from the sin(tx)/x integral. Since it's result is π/2 for any t value that means even 3t is a value that works. So t=1,2,'3',4,5,'6',7... work which includes the multiples of 3 ie. t=3,6,9...

@selectname9790 Жыл бұрын

@@noopcode but yeah I think he just missed writing the 3

@Manuel_Gestal Жыл бұрын

10:13 wouldn't it be sin(3tx) instead of sin(tx) ??

@boomgmr6403 Жыл бұрын

At 11:54 you dont write sin3tx again, is that a mistake?

@boomgmr6403 Жыл бұрын

dont you then get integral sin3tx/x dx? Does that change something?

@Gamedolf Жыл бұрын

At 10:50 he says you can have any constant multiple and it will always be pi/2

@djsmeguk Жыл бұрын

Yes, but also no. The result is always pi/2 so it's not significant.

@boomgmr6403 Жыл бұрын

@@Gamedolf I see

@MarkPaul1316 Жыл бұрын

@@boomgmr6403 he should have written it as sin3tx and shown that making the substitution u = 3tx would take the integral from 0 to infinity of (sinu)/u which gives pi/2.

@fantastic1046 Жыл бұрын

We can get the same result by a sampled function and sampled squared in frequency domain then taking area at freq = 0

@premdeepkhatri1441 Ай бұрын

Thank You for this video.

@AntimatterBeam8954 Жыл бұрын

I just bought calculus clothing off your store. My weird maths science wardrobe is increasing. I am happier than I was 30 mins ago now.

@melstadevosyan Жыл бұрын

How did sin(3tx) became sin(tx)? Are they equal?

@fadihamed4826 Жыл бұрын

I've the same question also ... that it makes my brain explode

@richardbraakman7469 Жыл бұрын

I think it was a mistake that didn't matter to the answer, because the integral of sin(ax)/x dx is the same for any nonzero a

@maalikserebryakov Жыл бұрын

you can use de moivre angular expansion to write trig^n as a sum of linear trig

@God-ld6ll Жыл бұрын

use infinity & beyond. works wonders

@scottleung9587 Жыл бұрын

Damn, that's hardcore - nice going!

@user-uv3tr1ei1b Жыл бұрын

bro u are insane pls make more viedos like this, i realy like it

@prollysine Жыл бұрын

Hi bprp, thank you, the complicated calculation can be followed. Yes, Mr. Feynman could not only joke... I will study a lot...

@josdurkstraful Жыл бұрын

I understood absolutely nothing of all this but still watched the whole video...

@syamantagogoi Күн бұрын

10:14 In the second integral it should have been Sin(3tx) in numerator and I think it would be difficult to get the final answer in this context.

@PunmasterSTP Жыл бұрын

That was some Feynmania!

@erictrefeu5041 Жыл бұрын

une formule générale pour l'intégrale de (sin(x)/x)^m: kzfaq.infoOadiTfmwjTI

@gooball2005 Жыл бұрын

I like to think that he's just in somebody else's office at 12:30 in the morning doing integrals

@scienceresearchwithishan6965 Жыл бұрын

Bro u should also make a video doing this from contour integration using residue theorem and Jordan's lemma 😁😁

@mikefochtman7164 5 ай бұрын

I lost something at 10:16. You got 9/4 int(0,inf) sin(tx)/x. But the previous step it was 3/4 sin(3tx)3x/x^2 ?? Shouldn't the argument to the sin function still be 3tx? How did that 3 disappear?

@abdulmalek1118 Жыл бұрын

I have found a pretty nice question that I will suggest Solve the system a = exp (a) . cos (b) b= exp (a) . sin (b) Can be solved easily using Lambert "W" function by computing ( a+ib ) And thanks

@Master_mind__235 Жыл бұрын

To take out √-1 put e^iπ in the place of -1 Please do it !

@rogerdudra178 Жыл бұрын

Greetings from the BIG SKY. Nothing like a bit of calc to end the day.

@imsengky Жыл бұрын

It is so good. I am really happy to see that solution. Thank

@dipankarbanerjee1130 Жыл бұрын

Actually what is Feynman's trick ? I am a high school student and this isn't in my syllabus but I am eager to know

@jatingupta6198 Жыл бұрын

I didn't understand what u did but i would have used product rule of integration using ILATE😅

@AbhishekSachans Жыл бұрын

At 10:20, in the sex nd integral, it should be sin(3t).

@holyshit922 Жыл бұрын

Integrate by parts twice with D I sin^3(x) 1/x^3 then substitute u=3x finally i got 3/4Int(sin(x)/x,x=0..infinity) then i calculated Laplace transform L(sin(t)/t) and plugged in s = 0

@armanavagyan1876 Жыл бұрын

I adore your videos i watched the half of 100 integrals)

@khoozu7802 Жыл бұрын

A fastest way is applying the formula sin^3(x)=1/4(3sinx-sin3x) And using integrate by parts

@ee-prakalyadav Жыл бұрын

I solve these question in my paper . But your techniques are quite impressive

@frederickwong4390 9 ай бұрын

I think using the triple angle formula sin(3x)=3sin(x)-4(sin(x))^3 is easier. Unlike what other have said, there is no need to use contour integration.

@danmart1879 Жыл бұрын

I was lost for most of the video !! I have a long way to go.

@darkknight32920 Жыл бұрын

Sorry for the naive question, but when solving for c, what if you let t equal any multiple of pi? Wouldn't that change what c is? Why is it possible to choose the "easiest" value?

@KingGisInDaHouse Жыл бұрын

Wouldn’t complex analysis work here?

@mcalkis5771 Жыл бұрын

After the 100-x series I am surprised you want to even SEE another integral ever again.

@boombam5589 Жыл бұрын

7:25 Evil laughter 🤣

@yabaminozomi 11 ай бұрын

10:14 why the sin (3tx) suddenly becomes sin(tx)

@Johnny-tw5pr Жыл бұрын

Would this work? I(t)=(integral)sin^tx/x^t

@aadisankar.s4449 Жыл бұрын

Sir, please explain why there exists two types of vector products...

@ES-qe1nh Жыл бұрын

There's three, actually

@lilfelicia Жыл бұрын

At 6:20 why do we have cos(tx)-cos(2tx)?Where does the minus come from? And why do we have a plus afterwards?

@uncelesteperro8258 Жыл бұрын

10:12 how did he get rid of the 3 on sine's angle?

@ayoubelouafy6174 Жыл бұрын

There's a mistake in the 2nd derivative of I(t) in the 2nd line you got sin(3tx) in the 2nd term. All respect to u it's a hard integral .

@Mini_Wolf. 2 ай бұрын

Can you do the taylor series of sin and work from there?

@jackychan4640 Жыл бұрын

我想祝福你新年快樂happy Lunar New Year

@saurabhkatiyar2704 Жыл бұрын

Very very easy question

@tahsintarif6864 Жыл бұрын

make a video on solving 100 Putnam Calc 2 Problems

@yokoyapen Жыл бұрын

9:32 the 2 appears like magic

@LuisHernandez-ip7gx Жыл бұрын

Muchas gracias

@manishkumardeep2230 Жыл бұрын

PLEASE MAKE A VIDEO ON FORBENIUS METHOD OF SPECIAL FUNCTIONS

@mrwest9840 Жыл бұрын

How can we write d/dx( sin^3x) = 3sin^2x cosx ?????? 😏

@francescorosatelli6001 Жыл бұрын

sin(3tx) right high corner , where did he go?

@bisheshshakya3838 Жыл бұрын

10:13 If I'm not mistaken, it's supposed to be sin(3tx) but you wrote sin(tx)....please clarify?

@nickharrison3748 Жыл бұрын

what is the practical use of this equation?

@arsh.008 Жыл бұрын

On the right half of the board, the second step where you took negative common, shouldn't it be -(9/4) and so on?

@richardbraakman7469 Жыл бұрын

No he split the expression at the +, so the left half is for -sin(tx)•x and the right half is for sin(3tx)•3x

@INSANITY335 Жыл бұрын

we can even use sin3theta here right?

@jeanmaxcoransoni2183 Жыл бұрын

At 10:14 : error sin(3tx) not sin(tx)

@alibekturashev6251 Жыл бұрын

i have never seen you that tired🥺

@chumdjr Жыл бұрын

請問曹老師，封面的獎牌是？（數奧？還是⋯）

@blackpenredpen Жыл бұрын

馬拉松獎牌因為這是我一次做一百題數學的影片片段

@user-mh3vr5rw9r Жыл бұрын

Can you solve the integral of : ln(sinx+cosx)/(cosx-sinx) dx

@krishnankuttyp4478 Жыл бұрын

Sin3xt/x put 3xt=u x=u/3t 3tdx=du dx=du/3t integral sin3xt/xdx=(sinu×du/3t)/u/3t =integral sinu/udu =pie/2

@SakshamSiwa Күн бұрын

Bro what is your educational qualifications?

@user-yc3fw6vq5n Жыл бұрын

Wait what? Feynman invented math? I thought Feynman was a physicist . . .

@nirvikthapa9436 Жыл бұрын

Me trying to figure out is it 12 Am or Pm in the clock

@maalikserebryakov Жыл бұрын

5:20 this is the main weakness in your approach ive noticed When you arrive at two viable techniques to modify the integration you force yourself to choose. Just do both!

@tarentinobg Жыл бұрын

I'd use brackets [ ] instead of multiple parentheses. Love your work. Thank you.

@ThreePointOneFou Жыл бұрын

So would I. Unfortunately, the old rule about surrounding parentheses with brackets (and brackets with braces) is becoming increasingly out of vogue. Current standards of mathematical notation lean toward using all parentheses, even though it can obscure nested expressions.

@krishnanadityan2017 5 ай бұрын

The second term in I'''(t) expression on RHS is not correct

@alexandermorozov2248 2 ай бұрын

10:13 - sin(3tx) !! 😜

@parhuzamosgyorgy5310 Жыл бұрын

With Fourier transform the difficulty level drops to ** max.

@romanbykov5922 Жыл бұрын

9:45 why did you differentiate the numerator here but didn't you differentiate the denominator of x^2?

@PaoloCasillo Жыл бұрын

Because x^2 is a constant in t world.

@GauravKumar-vl7rt Жыл бұрын

Love from India

@crescenzosimeolisimeoli8756 Жыл бұрын

Is this integral generalizable for n?

@ihatethesensors Жыл бұрын

I asked ChatGPT to do this and it got pi/2. It's 3pi/8, so I assume it actually got 4pi/8 originally. Very close. How do you suppose it got that? Wrong answers on the internet? Or did it actually try to follow the procedure and make a simple mistake at the end? Still, quite impressive.

@richardbraakman7469 Жыл бұрын

The answers with sin and with sin squared are pi/2, so it probably just picked those even though they don't apply for sin cubed