Evaluating challenging integrals via differentiation: Leibniz rule

[@DrChrisTisdell]

I would very much like to spend some time at Edinburgh University. It has a good reputation and a wonderful tradition. My Mum also happens to be Scottish-born (but in Glasgow). Glad to hear you're finding the vids useful. You may also like my new ebook - the link is in the description.

[@DrChrisTisdell]

You are right. In the sample paper, you start with a given identity and differentiate both sides, applying Leibniz' rule in the process. In this vid there is no 'given identity' (to differentiate both sides of). In addition, the solution in this video involves solving a simple differential equation, which is not seen in the solution to the sample paper question. I have posted another vid that is very similar to the sample test Leibniz question. Good luck!!

[@DrChrisTisdell]

Happy to know that the vids are begin watched and are useful!

[@DrChrisTisdell]

In the case of constants in the integral sign (like the one I did in this integral) Leibniz rule is not interesting or effective for integrands which are functions of one (dummy) variable. However, you can make some conclusions about integrals with integrands which are functions of one (dummy) variable. For example, in this video if you solve it and then find $I(2)$ then this obviously shows that $\int_0^1 (x^2 - 1)/(\ln x) dx = \ln 3$.

[@DrChrisTisdell]

@sat1354 Are you looking for a general antiderivative, or is your problem involving a definite integral?

[@DrChrisTisdell]

@MrEnigma786 I'm not quite sure of your question, but I'll try to answer: The (\alpha + 1) comes for the integration of x^\alpha from x=0 to x=1. If this isn't useful then pleaes give me some more info so that I can address your question more directly.

[@DrChrisTisdell]

That is a great question and the answer is yes. You can use the Chain Rule and FTC to obtain a more general Leibniz rule when differentiable functions appear in the limits of integration. If you would like to see the full result and a proof then I recommend you see p.738 of "Calculus. A Complete Course" (6th Ed) by Robert A Adams, Pearson, 2006. (There may also be online material!)

[@DrChrisTisdell]

Great comment about HD! I think HD is important so that the viewer can see all the little details of the solution. Details are so important in mathematics!

[@stepsvideos]

Thanks very much for the video. This was not taught when I was learning calculus (about 30 years ago). Had some fun using it with the CRC integral tables, finding solutions to integrals that are not listed, by taking the derivative of solutions to listed integrals. If that makes sense...

[@DrChrisTisdell]

Although people like Richard Feynmann (a physicist) have tried to popularize this technique, it is still relatively little-known. :-)

[@ravi00090]

Thank you sir. The way you teach is so clear and outright fantastic. I now understand Leibniz rule very very properly. The only problem with me is when I reach past a certain level of ease in the initial types of sums, I begin to get lost in figuring out the question mainly [simplifying and understanding the question]. I think that has got more to do with my very weak knowledge of Trigonometry. That makes me feel very nervous and scared.

[@sat1354]

Great explanation Chris!! Im actually trying to integrate just (x^2-1)/lnx now by fundamental techniques and dont really know if it is possible. What would you suggest?if you dont mind to help a fellow mathematician?)))Thanks for great videos again!!

[@شغف-ت6ع]

كيف عوضتت في التكامل وطلعت القيمة بكل سهولة ؟ بصير اعوض في التكامل مباشرة؟

[@DrChrisTisdell]

Best of luck for the test!

[@DrChrisTisdell]

@darkonica That's correct - there is a more general form (which can be derived via the Fundamental Theorem of Calculus and the Chain Rule). See Q48 or Q49 in /watch?v=VOowCWqQj_g to give you some idea of how to derive it.

[@sat1354]

@DrChrisTisdell Yes, im just looking for a general antiderivitive!

[@postbodzapism]

Why is I(alpha) defined? when x is zero, ln x is undefined and when x = 1, ln x = 0 and the denominator is 0.....

[@egor.okhterov]

If I knew that, I would have passed the entrance exam yesterday and became magister of science (well.. eventually). Now I know that technique and what should I do? :)

[@nikz113]

chris if only you were my maths professor life would be much easier...ever thought of teaching at edinburgh uni?

[@DrChrisTisdell]

Thank you!

[@Dariusstarkus]

Thanks mate, great explanation ;]

[@REFG89]

Cheers!

[@jalalson61]

17 years

[@Yu2Kal]

im not good enough at maths to do this haha (yet)

[@bmthnumber1]

Kid's stuff.