Leibniz' rule: Integration via differentiation under integral sign

I discuss and solve an example where, given a simpler integral, a more complicated integral is evaluated through differentiation. The method features an application of Leibniz' rule for differentiating an integral. Such an example is seen in 2nd-year university mathematics.

[wmd]

I dealt with this in calculus, and now Leibniz has come back to find me in philosophy class. This guy was EVERYWERE!

[DrChrisTisdell]

I'm glad you liked this one!

The general "Leibniz method" is very useful for evaluating extremely challenging integrals (much more difficult integrals than the one in this video),

[darkconjure]

Saves a lot of grief for some one reviewing this rule. Thx

[DrChrisTisdell]

Hi jslashs, I've been on holidays this week, but will try to post something tomorrow night so you have the weekend to look it over before the test.

[TheLeeMrLee]

thanks for this video- I'm an A-level student in England, going to uni for maths starting this september, and I've been trying to read around some more advanced material from the standard A-level syllabus and I came across this method of integration, and despite my best efforts, I could not understand it until I saw this, thanks a lot :D

[DrChrisTisdell]

Hi REFG89, I've been on holidays this week, but will try to post something tomorrow night so you have the weekend to look it over before the test. Make sure you study the sample tests that I posted on My eLearning Vista!

[frelsmeg]

Dr Tisdell,
What about the case when the integration variable and differentiation variable are the same ? For example, how would Leibniz's rule be applied for evaluating: dx_dt of the definite integral ?

[DrChrisTisdell]

theneltrajan: you're right, I don't use "voice modulation' in these clips like I do in class. However, if you check the latest "Lagrange multipliers" vid then at 0.52 and 2.00 there's just a hint of an accent!? :-)

[jackleong]

It's really helpful :)
and it's my tutorial question as well :)
just had my exam today :)

[SolidSnyder]

Thanks so much! I've been working on the first derivative of the gamma function today, and that helps a lot.

[DrChrisTisdell]

Always great to get your feedback! :-)

[DrChrisTisdell]

Thanks for watching!

Around the 25 second mark I mention how to evaluate the first integral (so that you can verify the expression for $I$ is correct).

Good luck!

[fcmitroi]

@DrChrisTisdell I teach and I am a PhD student in mathematics.Your videos are very interesting to me (I discovered them just few days ago), not only from math viewpoint, but also the style of your talks and the fluency of ideas and language.
Math history is quite interesting but there are too many Bernoulli there, we should count them like French people count their Louis (1st, 2nd, 3rd).
I saw Norman's name in your subscriptions list, so I intended to look at some of his videos. Thank you.

[REFG]

I agree with jslashs - leibniz's rule seems to be one of the more difficult concepts in the course and there are not enough examples in the lectures notes/course pack to be able to understand the variety of leibniz questions in the tutorial problems - some more videos would be great :)
Cheers

[DrChrisTisdell]

@fcmitroi This is a question that I pose each year in lectures to my students. It is an interesting story about his association with Bernoulli.

[himmatpanag]

Hey, do these videos cover all or more of the UNSW course Higher Several Variable Calculus MATH2111?

[DrChrisTisdell]

Great question Jedi! The new limits will be $0$ and $\pi/2$. Can you see why? (Hint: draw the graph of $tan x$ to gain some insight).

Good luck!

[DrChrisTisdell]

@mangymetal I believe that both variations are OK - I have seen both spellings of each name, however, Wiki uses the spelling "Leibniz". My understanding is that both my versions of L'Hop/L'Hosp are correct. Best wishes .

[jediwhelan]

Hi - great video, and i understand Leibniz's rule.

But can you help me with the substitution you refer to at the start please. I understand if you make the substitution

x = a * tan(t) then dx = a sec^2(t) dt

and after using the identity tan^2(t) + 1 = sec^2(t)

INT 1/a dt

right?

but how do i deal with the limits? Or have i gone about the substitution wrong. Many thanks for the help, i just lost an hour of my life!

[DrChrisTisdell]

@fcmitroi : ) The older I get, the more interested I become on math history. Do you know Norman Wildberger ( his channel is njwildberger )? He has an excellent playlist on the history of math.
You obviously have a very good knowledge of mathematics - are you a mathematician / teacher / researcher?