The Dirichlet Integral is destroyed by Feynman's Trick

The Dirichlet integral (integral from 0 to infinity of the sin(x)/x also know as the sinc function), is typically not taught in first year calculus courses. But the trick to solve it is actually pretty easy! In this video I show how we can use Feynman's Trick to make it a big messier by including a new exponential factor, but by differentiating under the integral sign the messy x in the denominator gets cleaned right up.

Check out my MATH MERCH line in collaboration with Beautiful Equations

COURSE PLAYLISTS:

OTHER PLAYLISTS:
► Learning Math Series
►Cool Math Series:

BECOME A MEMBER:

MATH BOOKS I LOVE (affilliate link):

SOCIALS:

[LaughingManRa]

Feynman DESTROYS Dirichlet Integral with FACTS and LOGIC

[kdmdlo]

Trefor: this trick requires that the order of differentiation and integration can be interchanged because your F'(s) is actually d/ds [ int_0^infty e^(-sx) sin(x)/x dx ] . You nonchalantly swapped the order of these two operations. This is thoroughly valid provided the integrand satisfies certain integrability conditions etc. Perhaps it would be worth noting this (and going over these conditions), so if people are looking to use Feynman's trick, they will be on the look out for potential tripping points.

[Sugarman]

Since you briefly mentioned the Laplace transform, I feel like it'd be a waste not to mention the super important Fourier transform in this context, because the Fourier transform lets you solve the Dirichlet integral almost immediately. It turns out, the Fourier transform of a window function of from -1 to 1 is sin(w)/w, so using the inverse fourier transform, you get the value of the Dirichlet integral.

[Djenzh]

Beautiful! Funny that the Laplace transform shows up. I only knew how to do this integral by changing sin(x)/x into sin(xt)/x and then taking the Laplace transform of the entire thing, but your solution seems much easier :)

[NumbToons]

I just (today) learnt this integral in Fourier Transform, and here you come up with a video to make it permanent my memory.

[emanuellandeholm]

I always love to watch different people's take on the Dirichlet integral. It's second only to the Gaussian integral for me. :)

The interesting thing about the Dirichlet integral is that it's not Lebesgue-integrable. Put some of that stuff in your pipe and smoke it!

Dr Bazett's take is basically a Laplace transform, and I think it's cute!

[cablethelarryguy]

You have no idea how many videos I've watched about Feynmann's technique. I finally understand it. Thank you so much!

[moritzberner]

I directly tried to solve the integral of cos(x)/x with the same method and found out that this one diverges. Great video!

[purplenanite]

That's a neat trick!
i came across this integral yesterday, interestingly enough - although now I know how to do it faster!

[ரக்ஷித்]

The beauty of Laplace transforms! Amazing video❤

[laydenhalcomb]

I always love your videos Dr. B. Thank you for sharing. I’m now going to utilize the Feynman trick in my calc 3 class

[scollyer.tuition]

The terminology related to this method is itself somewhat interesting - when I first came across it (about 40 years ago), I don't think it was given any specific name (except differentiating under the integral sign), then the name "Leibniz rule" seemed to become more popular, and in the last 5 years or so, the "Feynman method" began to reign supreme - a tribute to continuing popularity of Mr Feynman, I guess.

And it's worth pointing out that there are conditions required for the method to work: continuity of f(x, s) in both x and s and (partial) df/ds over the region of integration, IIRC. (corrections gratefully accepted if I misremember).

Also, there's a generalisation of the method that takes account of variable limits depending on s.

[flamitique]

Great video ! I just wanted to point out that even though the result is correct, the reasoning here wasn't completely true, or was at least incomplete : by this reasoning, which uses the dominated convergence theorem and it's equivalents to derivate under the integral, you can't directly prove this formula for all s greater or equal to zero, but the formula is only true for s stricly bigger than zero, meaning you can't directly plug in 0 at the end (this is because you can only dominate the first function you want to derive under the integral for all s>0). But the formula still holds for all s>0. So the correct way to prove it is to show that the limit as s goes to zero of F(s) is indeed the integral of sin(x)/x from 0 to infinity, and you can use the fact that the right side of the equation is continuous at zero to give the final result. However, showing that you can interchange the limit and the integral as s goes to zero is not that trivial, since you can't dominate the function properly. To do that, you first need to integrate by parts and only after that you can dominate the function properly and do an interchange of limits and integral that is valid, and get the final result.

[ketankyadar]

I am really happy to see Differentiation Under Integral sign rule here to calculate integration of Sinx/x.
Actually today in the class i taught this rule and after that i saw your video.
I amazed that how you start with combining exponential term in the integral, In real life there are so many situations where you can use this cause there exist always parameter with your function.

[giovanni]

You can't plug in s = 0 as the Feynman trick can only be applied with s > 0, due to the absolute convergence requirement, though the limit as s goes to 0 indeed equals pi/2

[andrewharrison]

Some bits of maths are like p v np, really hard to first find the method (np), but manageable to verify that the solution works (p). Actually it isn't "some bits", it is a lot of the bits and it is cumulative - Newton's "If I have seen further than other men it is by standing on the shoulders of giants".

[ShanBojack]

You remind me of my tuition teacher who is also a big mathemagician and you both are my ideals 🙌

[maroc]

Great explanation. My follow up question would be, why does this work and when should one use this trick? I only knew the Double integral solution for this Problem.

[ரக்ஷித்]

3:10 This can also be solved using complex numbers by rewriting this as Im(e^(-sx-ix)).

[johnchessant]

Fun fact: the integral of sin(x)^2/x^2 from 0 to infinity is also pi/2