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

Подписаться 1,3 млн

Просмотров 250 тыс.

50% 1

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 part 2 (...
🛍 Shop math t-shirts & hoodies: bit.ly/bprpmerch
10% off with the code "WELCOME10"
----------------------------------------
**Thanks to ALL my lovely patrons for supporting my channel and believing in what I do**
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N
----------------------------------------
💪 If you would also like to support this channel and have your name in the video description, then you could become my patron here / blackpenredpen

Опубликовано:

23 сен 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 210

@blackpenredpen Год назад

This is from the 100 integrals part 2. See the full video here ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-jQz1gQ24OHc.html

@lorenzosaudito Год назад

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

@wonghonkongjames4495 Год назад

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 Год назад

I Watch your video from India

@lordofhunger5175 Год назад

We need a tutorial about where to use each pen

@donkeybros8734 Год назад

He has a RU-vid short

@geneyoungdho Год назад

Characteristic part

@patrickinocencio2305 Год назад

agree

@mathssucker Год назад

hahaha I have a tutorial!

@Mehmetbilendegilypo Год назад

youtube.com/@mehmetturk6583

@weinihao3632 Год назад

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.

@maalikserebryakov Год назад

@julw9138 who asked?

@CurryMuncher2 8 месяцев назад

@@maalikserebryakovhuh?

@LuigiElettrico Год назад

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!

@yutaj5296 Год назад

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

@vatsalmalav440 Год назад

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.

@jamiewalker329 Год назад

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 Год назад

Indeed.

@chayanaggarwal3431 Год назад

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 Год назад

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

@CliffSedge-nu5fv 2 месяца назад

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

@drpeyam Год назад

Reminds me of the Borwein integrals a bit

@h10r60v 5 месяцев назад

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

@sngash Год назад

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

@zunaidparker Год назад

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.ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-851U557j6HE.html

@omograbi Год назад

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

@ianfowler9340 Год назад

I would say it's a different answer.

@MarkPaul1316 Год назад

@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 Год назад

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 Год назад

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

@krisbrandenberger544 Год назад

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 Год назад

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 Год назад

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 Год назад

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 Год назад

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 Год назад

@@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 Год назад

@@amirbasson532 ohhhh that explains it, thanks!

@-fai7485 Год назад

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.

@mumilala9940 Год назад

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

@md2perpe Год назад

I used that technique for the integral of sin²x/(x²(1+x²)), seen in ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-S52DapoH17M.html

@DanielCohen-d4v Год назад

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

@CDChester Год назад

Another integral for the collection!

@vogelvogeltje Год назад

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.

@abdulmalek1118 Год назад

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

@kelvin31272 19 дней назад

Email him!

@pacifyplayer 5 месяцев назад

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?

@kelvin31272 19 дней назад

So here's what I think: the missing 3 was a copying mistake, which luckily turns out not to have any consequence on the answer, since the definite integral of the form sin(ax)/x evaluated between 0 and inf, is always equal to π/2, for all a > 0 (I think!) That means no matter if it is sin(3t) or sin(t) on the top, both integrals are still of the form sin(ax)/x (assuming t> 0 so that a>0, which was indeed stated earlier), and so both integrals are still π/2. If you want to prove that this is true, solve by subbing u=ax, and thus dx=du/a, into the integral of sin(ax)/x evaluated between 0 and inf, simplify, and you'll see it becomes the integral of sin(u)/u from 0 to inf, which is already known to be π/2. Thus the integral in that line where he makes the mistake, with or without the 3, is still equivalent to π/2, and there is no impact on the final answer. I hope this helped!

@boomgmr6403 Год назад

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

@boomgmr6403 Год назад

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

@Gamedolf Год назад

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

@djsmeguk Год назад

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

@boomgmr6403 Год назад

@@Gamedolf I see

@MarkPaul1316 Год назад

@@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.

@wcottee Год назад

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

@ActALCOCERBONILLAARTUROAZAEL Год назад

Same doubt

@noopcode Год назад

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

@cristofer6806 Год назад

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

@selectname9790 Год назад

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 Год назад

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

@Manuel_Gestal Год назад

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

@fantastic1046 Год назад

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

@melstadevosyan Год назад

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

@fadihamed4826 Год назад

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

@richardbraakman7469 Год назад

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

@God-ld6ll Год назад

use infinity & beyond. works wonders

@premdeepkhatri1441 2 месяца назад

Thank You for this video.

@fordtimelord8673 Год назад

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. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-ovj71qp7C4k.html

@AntimatterBeam8954 Год назад

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.

@maalikserebryakov Год назад

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

@АлександрКузнецов-р6ю Год назад

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

@PunmasterSTP Год назад

That was some Feynmania!

@scottleung9587 Год назад

Damn, that's hardcore - nice going!

@prollysine Год назад

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

@Master_mind__235 Год назад

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

@imsengky Год назад

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

@scienceresearchwithishan6965 Год назад

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

@mikefochtman7164 6 месяцев назад

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?

@kelvin31272 19 дней назад

@AbhishekSachans Год назад

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

@holyshit922 Год назад

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 Год назад

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

@josdurkstraful Год назад

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

@gooball2005 Год назад

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

@syamantagogoi Месяц назад

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.

@yabaminozomi Год назад

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

@dipankarbanerjee1130 Год назад

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

@khoozu7802 Год назад

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

@abdulmalek1118 Год назад

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

@jatingupta6198 Год назад

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

@frederickwong4390 10 месяцев назад

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.

@aadisankar.s4449 Год назад

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

@ES-qe1nh Год назад

There's three, actually

@KingGisInDaHouse Год назад

Wouldn’t complex analysis work here?

@ee-prakalyadav Год назад

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

@lilfelicia Год назад

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?

@rogerdudra178 Год назад

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

@Mini_Wolf. 3 месяца назад

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

@darkknight32920 Год назад

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?

@uncelesteperro8258 Год назад

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

@EntropicNightmare Год назад

When you do the u substitution when computing I''(t), you get constant 3pi/4, which is fine for all t>0, but how do you justify that at t=0 where the integral appears to give zero when you go to fix your constants of integration?

@ayoubelouafy6174 Год назад

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 .

@danmart1879 Год назад

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

@bisheshshakya3838 Год назад

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

@jackychan4640 Год назад

我想祝福你新年快樂happy Lunar New Year

@mcalkis5771 Год назад

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

@yokoyapen Год назад

9:32 the 2 appears like magic

@manishkumardeep2230 Год назад

PLEASE MAKE A VIDEO ON FORBENIUS METHOD OF SPECIAL FUNCTIONS

@tahsintarif6864 Год назад

make a video on solving 100 Putnam Calc 2 Problems

@Johnny-tw5pr Год назад

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

@francescorosatelli6001 Год назад

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

@chumdjr Год назад

請問曹老師，封面的獎牌是？（數奧？還是⋯）

@blackpenredpen Год назад

馬拉松獎牌因為這是我一次做一百題數學的影片片段

@arsh.008 Год назад

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 Год назад

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 Год назад

we can even use sin3theta here right?

@boombam5589 Год назад

7:25 Evil laughter 🤣

@mrwest9840 Год назад

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

@nickharrison3748 Год назад

what is the practical use of this equation?

@jeanmaxcoransoni2183 Год назад

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

@nirvikthapa9436 Год назад

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

@LuisHernandez-ip7gx Год назад

Muchas gracias

@romanbykov5922 Год назад

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

@PaoloCasillo Год назад

Because x^2 is a constant in t world.

@غازيالغزوان Год назад

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

@krishnankuttyp4478 Год назад

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

@user-yc3fw6vq5n Год назад

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

@saurabhkatiyar2704 Год назад

Very very easy question

@SakshamSiwa Месяц назад

Bro what is your educational qualifications?

@tarentinobg Год назад

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

@ThreePointOneFou Год назад

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.

@crescenzosimeolisimeoli8756 Год назад

Is this integral generalizable for n?

@alankuo5579 Год назад

What about Nth power

@Wout680 Год назад

Hey blackpenredpen, I can't figure it out, but why is x^(1/log_b(x)) equal to b (the base of the logarithm)?

@spaghetti1383 Год назад

Assume that identity is true. Then take log base b on both sides. Each side simplifies to 1. Logarithms are increasing and 1=1 so the identity must be true.

@Wout680 Год назад

@@spaghetti1383 That was a very clear explanation, thank you :)

@ihatethesensors Год назад

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 Год назад

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

@wydadiyoun Год назад

10:55 proof pleaaaaaaaaaaaaaaaaaaaaaaaaaase! why it always give pi/2 with any constant???

@wydadiyoun Год назад

ok nevermind, I figured it out with my effort

@thelittlesillystar Год назад

what if d/dx((sinh(x*ln(x)))^(2x*cos(x)))

@maalikserebryakov Год назад

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!

@alibekturashev6251 Год назад

i have never seen you that tired🥺

@emmagutielmejor Год назад

Borwein integrals ?

@wuzhai2009 Месяц назад

So many boxes of whiteboard markers!

@krishnanadityan2017 6 месяцев назад

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

@fadiel-riachi6675 Год назад

I am confused by the evaluation of I'(0) and I(0) at 12:14 and 13:26 respectively. Shouldn't (sin(x)/x)^n be equal to 1 at x=0 for positive integers n? Is there something I am missing? In order to have a nice value for I'(t), we need cos(tx) to be 0. In other words, we need to evaluate at tx= pi(k-1)/2, which changes the expression a lot and affects the next integration that finds I(t).

@richardbraakman7469 Год назад

If t = 0 then the sin(tx) term is 0 and the whole expression goes to 0. Remember that I is a function over t not over x, so x = 0 doesn't need to be considered anywhere.

@fadiel-riachi6675 Год назад

@@richardbraakman7469 Right, of course! Thank you!