Integration by parts (visualised)
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- Опубликовано: 10 июл 2024
- Integration by parts is one of the most useful tools for finding integrals! In this video we cover what it is, how it works and we also show what is VISUALLY happening!
Integration by parts is the product rule for Integration. It allows us to find the integral of functions multiplied together and it is a super interesting solution to a geometrical problem!
Problem Sheet: mathacy.com/problem-sheets
Timestamps:
0:00 - Intro
1:17 - Visualising integration by parts
4:42 - Examples
6:23 - Repeated/Nested Integration
7:05 - Integrating ln(x) (with unity)
8:00 - LIATE rule
Wow the animations are incredible! Keep up the great work!
Incredible video Mathacy! So well explained with the visualisation and the fantastic animations! ⭐️
Have been looking for a good visualization of integration by parts, glad to have finally found it. Good job!
Never thought of integration by parts with areas like this. I’ve seen it for derivatives however.😃
These animations are absolutely next level 🙏🙏🙏
Absolutely spectacular and still so underrated. How come I have no clue. Deserves millions.
that's one beautiful gem ❤ this is the first time I ever understand the logic behind integration by parts
Absolutely love that intro !!!
Amazing video, thank you so much for the clear explanation!
Your channel will be one of the most popular educational channels. Carry on. We need more video
Brilliant once again! Well explained 😃👏👏👏❤️
Thank you so much for the brilliant explanation.
very cool! and even with a problem sheet, thank you!
Calculus blesses you sir for this amazing lecture
This was great, high quality material.
4:15 are you sure you didn't mix the order up? if x = g(t) and dy = f'(t) dt then shouldn't the first substitution be the integral from a to b of g(t)f'(t)dt??
I've never heard of the "LIATE" mnemonic. Cool!
This is so helpful thank you so much
well made video, thank you!
Brilliant explanation! TY
Very good stuff, thanks
Developing a geometric intuition of calculus is very powerful.
Amazing!
I think at 4:15 you got your substitutions backwards (LHS should have become what the RHS became and vise versa)
Yes I think so too. Was wondering how that equation came to be 😅
Excellent explanation 👌 thanks
Awesome 👍🏼
Maaaaaannnnnnnnn..... This is what I neeeeeeddddd.....✨
Such an important video.
Good idea !
Thank u
Maaaaannnn.... You are great....!!!!!!🔥🔥🔥🔥
Beautiful Video
👍🏻👍🏻👍🏻thank you 👍🏻
It's amazing that the link in the description has free problems + the answers
Also, are the animations made with Manim?
Amazing video
really sad that u dont get that much views u put alote of effort in a good content and at the end this what u get :( shame on u youtube
best one !
0:19 The US should have better public transportation!!!! The "stress" someone can choose to feel because of something like this is crazy! Without a car one doesn't get to work!
It is "downright" "dangerous" to be a "pedestrian" in the US and most places do not have P.T.
Also, because there is no P.T. many kids are stuck in "house arrest" and "licenses" are given out like candy to people who probably should not be driving because of choices they make or "conditions" they might have!
Older people or people with high blood pressure don't go places and are more likely to die because of it (for example, exercise and other life choices can pretty much almost "eliminate" the chance of heart disease!)
Here in India they teach it as ILATE (inverse trigonometry first then logarithm)
What I didn't really understand was relating the examples with the visual parameterisation. For example when we took integral( -x sin x), what are the two parts of the graph that would be parameterized?
I wasn't able to relate exactly the examples with the basic visualization shown before.
Can someone share their views on this?
Could we say that f= x and g= cos x? but it doesn't really graphically make sense
I'm 11 months late, but if it helps, -xsin(x) is just a function in that example. It is not a parametric equation, otherwise we would need to know what x(t) or y(t) is like in the first example.
You use Manim to animate?
Ok, but what if you can't write x as a function of y? Why does integration by parts still work?
Is there a reason why LIATE is the best order to go by when substituting for f?
Try a different order, and then try with LIATE, and you will see which is easier.
Absolutely! The reason for Integrating by parts is to make a ‘product’ easier to handle.
In particular, the formula for integration by parts allows us to convert our integral into another integral where ONE of the functions is differentiated and the other is integrated (plus some other stuff we don’t need to worry about in the integral). This new integral should be easier to integrate otherwise our efforts would be wasted.
So, we want to know which functions we should differentiate to give us an integral that we can easily integrate.
It helps to work backwards here.
Exponential (E) is not useful to differentiate as its derivative is itself, so we’ll end up with the same integral.
Trigonometry (T) is not entirely useful because the derivative of trig functions usually gives us another trig function, which can be useful for some cancellations, but not the best in every situation.
Now Algebraic (A) is great as we can differentiate it quite easily, but if we have log(x) or arcsin(x) in our product it won’t help us as these functions will be in our new integral and we don’t know how to deal with them.
Now Logarithms (L) and Inverse Trig (I) can be used interchangeably as they are first for the same reason. For the finale, Differentiating these functions gives us algebraic functions! Which is fantastic as we know how to deal with these for the new integral. This converts a Logarithmic or Inverse Trig integral into a purely algebraic one. Check out our differentiation playlist / problem sheets if you are not sure how we differentiate these functions.
Great Question, hope this helps!
@@Mathacy Thanks for the explanation!
Why can't we take ILATE rule instead of LIATE
Hell yea calculus
so no one gon talk about how it was supposed to be a car repair video? XD
Watching this after final exam
300th like
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Good video, but why are you doing that with your voice?
Notations f(x) and g(y) in your explanations are incorrect. I think one should write y(x) and x(y).
Terrible
I tried to learn this, but it never proved useful to me. This bulk of it remains as gobbledegook. The information retrieved is meaningless
Yes why have a basis for anything?! Also I'm eager to hear you define gobbledegook!