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

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...
Zachary's page: philosophicalm... ,
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 ...
Subscribe for more math for fun videos 👉 bit.ly/3o2fMNo
For more calculus tutorials, check out my new channel @just calculus
👉 / justcalculus

[@112BALAGE112]

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

[@blackpenredpen]

hahahahaha! honestly, that wasn't planned.

[@ruiyingwu893]

blackpenredpen I just realised after reading this...

[@yamenarhim9336]

me 2 lollll

[@edwardtang3585]

It seemed to me like some sort of band sign like Nike at first

[@AlgyCuber]

what’s the difference between partial derivative and normal derivative?

[@terapode]

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

[@gertwallen]

Yes, Feynman was a brilliant mind

[@lisalisa9706]

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?

[@brandong5687]

Dokuta Viktor trust no one

[@arthurreitz9540]

Dokuta Viktor Ask wolfram.

[@blackpenredpen]

Dokuta Viktor only if it gives the same answer as what we got.

[@axemenace6637]

blackpenredpen what if what you got is by looking at Wolfram????

[@MingruiCHENG]

then don't get things from Wolfram but just check your answer with it.

[@whiz8569]

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.

[@antonquirgst2812]

its not that unexpected though if you look at the function... its just looks very convergent.. (this can ofc be very deceiving)

[@createyourownfuture5410]

@@antonquirgst2812 But there's the fact that as x grows larger, it tends to 0 because sin's at most 1 or -1.

[@antonquirgst2812]

@@createyourownfuture5410 yup - totally agree - x grows linear while sin(x) is periodic!

[@createyourownfuture5410]

@@antonquirgst2812 Aaaand it approaches 0 from both sides

[@josephcamavinga9721]

@@createyourownfuture5410 It actually approaches 1 from 0

[@siguardvolsung]

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

[@dannygjk]

lol

[@justinscheidler5938]

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.

[@jemcel0397]

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

[@blackpenredpen]

yay!!!!

[@MrAssassins117]

Yes, i did It and i got 10 in my integral calculus exam :') two months ago !

[@pranav2119]

@@MrAssassins117 now 3 years ago lol

[@_.Infinity._]

@@pranav2119 now 3 yrs and 14 hrs ago.

[@bonbonpony]

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

[@camilincamilero]

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.

[@taraspokalchuk7256]

this result is used to prove the convergent of fourier series though

[@PackSciences]

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.

[@footskills4953]

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

[@blackpenredpen]

Footskills here's the man!!!

[@Cannongabang]

Yeah that was a brief explanation haahahhahaha

[@footskills4953]

And here I am again!!! Btw, great explanation!

[@Tyns19]

PackSciences your counter example should be rearranged as (e^(-b x)-1)/x Btw e^(-b x)/x diverges for all values of "b"

[@modenaboy]

Can you like a video twice? Just watched this again, and still awesome. Thanks for this!

[@franciscoabusleme9085]

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

[@Aramil4]

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.

[@lantonovbg]

In fact, the integral from minus infinity to infinity of sin(x)/x IS equal to Pi. It is called Dirichlet integral. Thanks, ChatGPT

[@qbtc]

This technique is called "differentiating under the integral sign" and Feynman learned it from a book entitled Calculus For the Practical Man when he was a teen. Feynman didn't invent it but made it famous through his anecdotes.

[@ozzyfromspace]

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 🙌🏽🎊

[@jamesbentonticer4706]

One of the best videos on this great channel. Beautiful.

[@suhaimimazed1136]

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

[@blackpenredpen]

Mazed Suhaimi yay!!!!

[@sharpnova2]

at 20:23, the integrand is actually equal to a constant, not zero but that doesn't change the answer because you can combine arbitrary constants.. arbitrarily. 😋 and end up with the exact same answer

[@NoActuallyGo-KCUF-Yourself]

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.

[@venkybabu8140]

Write as 1/sinx/x . Expand 1/x^ 2 series and use Pi/2.

[@michaelmello42]

Inspired! Love this channel.

[@Barpoint212]

I love your enthusiasm and your clear explanations. Thanks!

[@lambda653]

You technically are supposed to take the limit of I(b) as b approaches 0 because I(b)'s boundary space is x>0, not equal or greater than 0. It's fine though because the limit as x approaches 0 of arctan(x) and arctan(0) are luckily the same.

[@AbdulWaris-ul3zx]

There are several methods to evaluate such integrals.It is based on Leibnitz theorem .

[@亨亨-y5v]

K 歌之王?

[@blackpenredpen]

Isaac TNT yes

[@__-rs8kr]

blackpenredpen man of culture

[@wontpower]

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

[@blackpenredpen]

I did, isn't it! hehehe

[@SanjeevKumar-js4mu]

@@blackpenredpen didn't I

[@adi-sngh]

@@SanjeevKumar-js4mu lol

[@randompasserby4575]

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)

[@thomasblackwell9507]

Would you please do a video on the derivation of this method and when you would use this method. Thank you.

[@edwardtang3585]

I love the piano cover of that Eason Chan song in the beginning!! Great taste of music

[@blackpenredpen]

Edward Tang thanks!!!

[@user-wf3oj2vv2w]

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

[@juanperez-ventana5621]

what a cool way to do the integral, thank you

[@ribozyme2899]

Nothing to check by derivation here, but it's definitely extremely cool!

[@blackpenredpen]

Ribozyme I will get 0 if I differentiate my answer.

[@JamalAhmadMalik]

Most satisfying video ever! Love you BlackpenRedpen!

[@executorarktanis2323]

13:19 that is cool

[@nmaedu.100]

Great Content of Mathematics

[@adrianstarly1]

what the hell, who would have guessed that complex analysis does actually make this integral way easier to calculate

[@sirjoseo.nastik2315]

14:03 : when you blink during the lecture

[@k_wl]

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

[@k_wl]

ok nvm some things change but i end up getting pi/2 answer anyway so ig it can be it

[@RobertGabor]

I like Asics because of elan extra speed ds/dt added while running.

[@chauokimix5361]

Thank you for this explains

[@arturaskarbocius6091]

This one Basel problem solution solved by Euler where PIE pops from nowhere :), and other method is Fourier series.

[@xoppa09]

i love this guy

[@NoNameAtAll2]

Red T-shirt? Are you Tom Scott?

[@blackpenredpen]

NoName who's that?

[@NoNameAtAll2]

blackpenredpen Really good youtuber Talks about interesting places to visit and interesting concepts to discuss

[@blackpenredpen]

does he always wear red tee?

[@haarithio1621]

blackpenredpen yep

[@blackpenredpen]

Ah! I see

[@leeyc0]

Is that only me noticed that the song at the beginning is a Hong Kong pop song K歌之王, sang by Eason Chan?

[@chiraldude]

I want his magic white board that suddenly shows the solution to the problem before he is done solving it!

[@timurrahmatulin1644]

14:02 like "Landau and Lifshitz", in smth moments - "It is obvious"

[@mrKreuzfeld]

When you let b go to infinity, it is not obvious that you can put the limit inside the integral. This is a job for lebesgue

[@poutineausyropderable7108]

With the e^-infinity integral part at 20:40, since the integral goes from 0 to infinity, wouldn't we have a 0*infinity limit problem? Since an integral is basically a better multiplication. Ah, oh, nevermind, the over x is going to kill it for high number an e^-infinity for lower number.

[@KK-wk1id]

That was the question in my exam.

[@YodaWhat]

Interesting... but of course, *one must know _when and where_ it is true that _each step is valid_ if one is to apply the technique more generally. Which makes me wonder: How would 3Blue1Brown explain this?

[@ESOMNOFUONLINEMATH]

Great job

[@_DD_15]

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

[@mohamedabdullahi5708]

Kkkkk

[@sharmisthaghosh9017]

Lol

[@camkiranratna]

For some reason , “lol” looks like mod(0)

[@deltaspace0]

@@camkiranratna do you mean absolute value?

[@dclrk8331]

@@deltaspace0 Absolute value is also called mod in some places.

[@AmanteNoViolao]

When you sleep in class 14:01

[@bonbonpony]

More like when you blink in class :)

[@peppybocan]

but the answer was spoiled in that part :D

[@Tomaplen]

when you struggle not to sleep

[@AhnafAbdullah]

Idk why was the video cut? lol

[@blackpenredpen]

Ahnaf Abdullah I wanted to add that explanation why b has to be nonnegative

[@sonicpawnsyou]

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

[@blackpenredpen]

MeowGrump lolllll this is a good one!!!

[@ffggddss]

asics = "Anime sane in corpore sano," "Sound mind/spirit in a sound body."

[@koharaisevo3666]

Anima not anime (but that's somehow relevant :))))

[@omarathon5922]

👌 looks like the partial derivative sign XD

[@herbert164]

So, it is soul eater then?

[@cycklist]

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

[@blackpenredpen]

PompeyDB it is!

[@jirehchoo2151]

it is, is not? It's!

[@rehmmyteon5016]

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

[@tens0r884]

@@rehmmyteon5016 lmao

[@rudycummings4671]

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.

[@gertwallen]

I agree, I solved this too in my first course of Applied Mathematics in college where we used complex analysis techniques kzbin.info/www/bejne/fJeXfYWim9GcmJI

[@greatwhitesufi]

Yeah that's true, that's how I learnt it/saw it first

[@lasmatesdelamor4287]

Integrales cerradas en variable compleja?

[@louisrobitaille5810]

You can do integrals on complex bounds (lower/upper) 😮? Or is it Real bounds but integrated on Complex functions?

[@comp.lex4]

@@louisrobitaille5810 complex functions and complex bounds. Turns out that the path you take *mostly* doesn't matter!

[@rishavmukherjee4251]

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

[@Lofila999]

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

[@not_vinkami]

……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

[@blackpenredpen]

Of course I know 😆

[@2070user]

ahhh, that's why the intro song is so familiar, k歌之王 by Eason Chan!

[@ShotgunLlama]

He's becoming self aware

[@Aramil4]

How so? What did you notice?

[@Reluxthelegend]

isn't it?

[@andraspongracz5996]

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!

[@hyungmanpark4346]

.l

[@seanclough7810]

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.

[@blackpenredpen]

Sean Clough yay! I am happy to hear!

[@MagnusSkiptonLLC]

Who else reads his shirt as "partial asics"?

[@mathnezmike]

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!

[@NazriB]

Lies again? So fat

[@samiali2434]

I came in just because i saw the name Feynman

[@abdullahbinjahed6900]

me too

[@supriya1423]

You got it ,me too

[@charliearcaro208]

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

[@Sugarman96]

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.

[@charliearcaro208]

@@Sugarman96 - the Laplace transform is what the above video uses when creating his function I (b)

[@mrocto329]

​@@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.

[@proofofalifetime488]

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!

[@nk4634]

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

[@yuchenwang679]

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?

[@MsMaciekk]

I think you're right. I was thinking the same

[@andreisecuiu6491]

Does it help? I am not an expert in the field (yet): en.wikipedia.org/wiki/Leibniz_integral_rule

[@WildSeven19]

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

[@Zonnymaka]

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!

[@michaeljohnston3038]

That's what she said

[@bigjosh2517]

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

[@NoActuallyGo-KCUF-Yourself]

Hard to justify with those zero to infy limits. ;-)

[@AndDiracisHisProphet]

so, pi/2 \approx inf?

[@kikones34]

How can you pretend all angles are small? The angle goes to infinity o_O

[@mike4ty4]

@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.

[@kikones34]

@mike4ty4 Oh, sorry, I totally didn't get you were joking. I've been on a KZbin 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:

[@sharmisthaghosh9017]

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

[@הראלישי-ר1מ]

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?

[@fabianpascalabt6353]

x approaches 0 from the right. With a weapon. Also discussed later in the comments

[@JoseDiaz-gp1bn]

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.

[@stephenmontes349]

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

[@blackpenredpen]

Paul Montes dr. Peyam is actually going to do that soon

[@FilipeOliveira-ir1hb]

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.

[@FilipeOliveira-ir1hb]

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.

[@me_hanics]

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

[@abedbob4046]

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.

[@benjaminbrady2385]

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

[@MrQwefty]

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

[@nikitakipriyanov7260]

You have to admit you've never solved differential equations before. This is very usual and boring thing, finding out these constants.

[@nathsmath2871]

00:00-00:23 K歌之王 :D

[@blackpenredpen]

LoSir MATH yea! I was waiting for someone to comment on that hahaha

[@穿第]

@@blackpenredpen 你係香港人?

[@鄧啃]

穿第 佢應該係大陸人，生活在國外（在美國？）的數學髙手。

[@generalsnicky3219]

@@穿第 可能是台湾人

[@alifa8903]

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.

[@S1nwar]

the world needs more of this....

[@mohanadou]

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 ?

[@ClumpypooCP]

Lmao the "isn't it" in the thumbnail

[@nin10dorox]

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.

[@antred9157]

I'm late, but I'll try to explain (sorry I'm french). Here you're using a theorem (with hypothesis he's isn't precising) to be able to differentiate in the integral. This theorem also tells u thats I and I' are continuous on there interval of definition. This means that ye, b>0 for I' but when you get the expression of I, since I is continuous on [0,+inf[, u can say that I(0)=lim I(b) when b->0 and thats it

[@davidkeys3962]

@@antred9157 Yes, b must be positive for the calculation. And one still needs to show that the function I is continuous at 0. This is in fact not easy to do, since the absolute value |sin(x)/x| is not integrable on [0,+inf]. One usually proceeds as follows: Instead of integrating over [0,+inf], define I_n(b), for positive b, to be the integral over [0,n]. Now use Feynman's trick. Proceed as in the video, and finally show that I_n(n) goes to 0 as n goes to infinity to complete the proof.

[@bruno-tt]

Beautiful proof, thank you.

[@restitutororbis964]

bruno edwards Yup, leibniz rule is very powerful.

[@icenarsin5283]

Best math teacher ever !!!

[@louisrobitaille5810]

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

[@EpicUnderscoreJdog]

just noticed that his shirt is the partial derivative of asics

[@roy1660]

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

[@thomasg6830]

The cut at 14:02 is kind of confusing.

[@dhvsheabdh]

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.

[@xxsamperrinxx3993]

It's so he can outline that b has to be positive, and it probably makes the most sense to put the clip here

[@adi-sngh]

When u sleep on class

[@user-en5vj6vr2u]

it spoiled the rest of the video

[@kakan147]

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

[@blackpenredpen]

Wonderful!!!!!!!!

[@markigolnikov6175]

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

[@usdescartes]

You can recover it, however, with just a little more rigor. Instead of evaluating I(0), find limb->0 I(b). Then, just keep using the limit notation until the end. The original integral is actually equal to limb->0 I(b) = Pi/2, so no real harm done.

[@aintaintaword666]

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

[@tharunmahesh7279]

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?

[@riccardopuca9310]

I also had this question. Anyone can help?

[@asirpagabriella5327]

@@riccardopuca9310 Maybe you have to set b>0, but when going back to the original, you let b approaches 0+?

[@stephenchurch1784]

The last step where he solves I(b) for b = 0 is a clever trick to avoid putting 0 into e^-bx. If you've taken diffeq, you can confirm for yourself by solving the original integral with a laplace transform. It'll also answer where the e^-bx came from in the first place

[@kalimachios]

had the same problem - i guess one can make b > 0, and then take the lim as b -> 0 from above on the I'(b) or I(b).. and would still be fine .. but how is presented, has that small issue

[@nostradamus9132]

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

[@damianmatma708]

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: kzbin.info/www/bejne/o2HSZ6N3mqiWgNU "Lobachevsky's integral formula and a nice application." Michael Penn