Feynman's Technique: This is the greatest integration method of All Time

Differentiation under the integral sign (Feynman's Technique) is such a useful tool, which pops up in many areas of maths and physics, that I had to do a video on it. I haven't introduced any rigour yet, but I'm happy to answer any questions in the comments.

[henryginn]

This is a cool example for those who know the technique already, but for someone who doesn't know it, I can't say I would recommend this video to learn it. For that, it would need to be longer, more motivated, and better explained

[Generalth]

This explanation is... honestly a little too simplified. It might have been a good idea to go into a _lot_ more detail with more steps, more examples, and more reasoning for when this technique can and can't be done.

[bernhardriemann]

Gottfried Liebniz knew about this technique 300 years before Feynman

[domc]

You need to mention the fact there's a +C . This is only the answer because I(0) = 0 so theres no added constant. This is a very important detail that you missed out

[antoniomantovani]

First of all F(x, t) must be definite in a set A measurable.
Then in A F(., t) must be integrable.
And last F(x, .) must be derivable and the derivate and < g(x)

[Metalhammer]

A bit more explanation and more examples would be nice. I mean, I know the trick/technique and just clicked out of curiosity, but yeah just a bit more in general would be helpful. Like if your channel is more for beginners, show the motivation, (which is simple, a lot of integrals are nasty buggers. Especially ones with denominators. And Feynam's technique is often godsent in taking those out) and a word of warning that you're not actually always allowed to switch the integral operator and the differential operator would be nice.

[debasishraychawdhuri]

It should work with any linear transformation, then integration and the anti-transform. Fourier transform, Laplace transform, Differentiation, etc.

[jamesexplainsmath]

Thank you everybody for all of your support and constructive criticism! Honestly, I didn't expect the video to get so many views. I made it for an audience of 100 or so people who already had plenty of mathematical knowledge as more of a refresher (which is why the video is WAY TOO concise). My content from now on will certainly be more explanatory! :D

I'll try and answer some commonly asked questions here:
- Q: 'Why is k=7?'
- A: 'I am answering the integral in the thumbnail and you should try k=2, 3, 4, ... as it is just an arbitrary constant.'

- Q: 'Why didn't you refer to Leibniz?'
- A: 'Feynman popularised the Leibniz integral rule in physics (pilot-wave theory is my honours topic, so I'm biased). Since Leibniz generalises for non-constant limits of integration, if I ever want to fix this video in the future, I will probably do the more general Leibniz integral rule.'

- Q: 'When can I swap the differential and integral signs? There is no rigour!'
- A: 'To differentiate under the integral sign, the integrand must be continuously differentiable (luckily most functions in physics are analytic and so this isn't much of a worry). To prove this, some real analysis is involved which I'm happy to go through in the future.'

- Q: Why didn't you take the indefinite integral instead of an integral from 0 to 7. Why haven't you mentioned +C?
- A: The infamous +C has tripped many a mathematician up on their final exam (including me lol). To be honest, it was a shortcut. I(k)=ln(k+1)+C, but by observation, I(0)=0 (using the definition of I(k)) and therefore C=0 -> I(7)=ln(8).

- Q: Why is a partial derivative of the LHS?

Anyways, if you are at the end of this, I hope this helps.

[adamwho]

This is a TERRIBLE explanation. Your second step don't follow from your first without several unstated assumptions.

You didn't explain any general principle.

[raunakmukherjee]

honestly, the most helpful video ive found on this. short and simple

[karldavis]

This video let me know, in 67 seconds, that the technique exists. To actually learn it, I would need something much more extensive, but that's fine, knowing it exists is a great start, and if the video were 20 minutes long and included practice problems (like needed to learn the method) I wouldn't have watched it.

[redbubble]

I really appreciated this short but precious piece of knowledge about calculus, thanks!

[dlive]

1:06 for Feynman's Technique? Not even Feynman would have explained it that fast.

[GiornoYoshikage]

Substitute ln(x) = -t, then the general integral will be from 0 to +∞ of (e^(-t) - e^(-(k+1)t))/t - a Frullani integral which is equal to ln((k+1)/1) = ln(k+1)

[christianartmann]

A typical video how math should not be explained i.e. leaving out all the necessary steps

[bobbybannerjee]

Very powerful. Almost shocking how simple the integral becomes which seemed hopeless on first sight!

[JunaidHasan]

There is a slight addition: When evaluating I(7) you have set the lower limit to be 0, but that may not always be the case. In general we set the lower limit to be that value where we know the value of the integral. In this case luckily I(0) = 0 and we are saved.

[otaku-qidj]

This video is the reason why YouTube should not remove the dislike button

[BorisNVM]

I did know the Feynman's tecnique, but I didn't notice that we could use it here. Nice one

[urlkrueger]

I think the first question you need to ask yourself is "Who is my intended audience?"
If it is first year calculus students or just people with an interest in math but limited coursework then you would need to provide more explanation and details.
If it is someone who has more advanced math skills but may not be familiar with this technique then they might appreciate your keeping it short as you did.