The Integral of sin²(x) Using Integration by Parts!

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The Integral of sin²(x) Using Integration by Parts!

Рет қаралды 16,651

Күн бұрын

Train your Integration skills and learn how to deal with Integration by Parts by trying out Brilliant! =D brilliant.org/...
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Today we evaluate sin^2 and cos^2 integrals using IBP. Some people like to call this technique "Phoenix Integration" lol Enjoy :)
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Пікірлер: 98

@PapaFlammy69 2 жыл бұрын

*_Hey folks, hope this one was more to your liking and you learned something new! :D If you feel like supporting the channel a bit more, why not check out some of today's relevant links?_* Check out my newest video over on @Flammy's Wood ! It turned out really nicely! =) kzfaq.info/get/bejne/m9agY7GcmL_Zo3U.html Train your Integration skills and learn how to deal with Integration by Parts by trying out Brilliant! =D brilliant.org/FlammableMaths Handcrafted products, puzzles and more :0 stemerch.eu/products/premium-end-grain-olive-wood-cutting-board?_pos=1&_sid=be52d5138&_ss=r ¬eULer'S iDeNTiTy Merch! :D stemerch.eu/collections/eulers-identity cos² No IBP Video: kzfaq.info/get/bejne/l9x4mLCas5rRlKs.html n-th Powers of cos and sin integral: kzfaq.info/get/bejne/Y7KSabugsZ-YaYU.html

@oni8337 2 жыл бұрын

😳

@HAL-oj4jb 2 жыл бұрын

As a physicist, I can say that the most common method for solving these integrals is to just look it up

@Christina.Anne. 2 жыл бұрын

Well, where’s the fun in that! 😂

@zyansheep 2 жыл бұрын

@@Christina.Anne. gotta go fast

@judedavis92 2 жыл бұрын

What do you work in physics? Particle physics? Nuclear?

@HAL-oj4jb 2 жыл бұрын

@@judedavis92 Astrophysics, solar physics to be precise :)

@pierfrancescopeperoni 2 жыл бұрын

@@Christina.Anne. Fun is in focusing on more interesting problems. Calculations like this are way too repetitive.

@peterschmidts8245 2 жыл бұрын

"You dont need trigonometric identities" proceeds to use the trig identity of sin^2 + cos^2 = 1

@KingGisInDaHouse 2 жыл бұрын

That can be easily argued by the Pythagorean theorem without knowing it though.

@1234larry1 2 ай бұрын

He’s talking about reduction formulas, so you memorize a formula and plug in numbers to get your answer. It doesn’t teach anyone math, just practicing arithmetic.

@stephendonovan9084 2 жыл бұрын

Hey Jens, sorry if the American comments on the last video felt like they were "blowing your ass up," I personally was just trying to let you know how it was done over here as a kind of cultural exchange, not as a knock to the video. My apologies if you felt harassed as a result. Take care man Edit: Something interesting I found about this method (although maybe it's old news to you) is that with a generalization of this method you can end up with a recurrences for antiderivatives for any even power of sine or cosine. In retrospect it seems obvious, but I thought it was worth mentioning.

@johnpesz6373 2 жыл бұрын

I love watching your integral related videos. You should do short video series called "an integral a day keeps the dementia away". It would be very popular in my opinion.

@Muzzycal 2 жыл бұрын

The first method i was ever shown was IBP too

@PapaFlammy69 2 жыл бұрын

good.

@toasticide816 2 жыл бұрын

Huh I mean I was taught to use integration by parts whenever there was a product as a first attempt. love the video! (Bri'ish by the way)

@That_One_Guy... 2 жыл бұрын

sin^2(x) = sin(x) * sin(x) so you can use IBP (where D and I are both sine) for this one

@That_One_Guy... 2 жыл бұрын

Even though that wasnt the usual method in my country too, it seems intuitive thing to do if you don't want to remember/forgot the trig identity (the cos(2x) one)

@Christina.Anne. 2 жыл бұрын

Thank you for another wonderful video! I love your Euler’s identity shirt, Papa Flammy! Euler’s formula is my favorite! I love studying it from an ontological perspective. 🥺

@PapaFlammy69 2 жыл бұрын

@hemandy94 2 жыл бұрын

In the curriculum where I teach, we teach integrating using identities first then IBP later so whenever I see something like this, I tend to use identities first. Nice to see it's different in other countries.

@dimitris713 2 жыл бұрын

Once i saw a really cool proof for the cos(2x) formula using linear algebra and i remember it really easily now , so i use this instead of IBP

@bhargav7215 2 жыл бұрын

Tbh, I've liked the video even before going Through it (dread confidence on Papa's content), but the moment I've watched it today has really made my day.thanks for the content Papa, as always "U r a man of intuition"

@PapaFlammy69 2 жыл бұрын

@mastershooter64 2 жыл бұрын

2:10 sadly that's how most stuff in India is. you're expected to just remember all the different formulas and most times they don't even show you the proofs for those formulas, so you can't even derive it, you have to remember it or look up the proofs.

@denizgoksu9868 2 жыл бұрын

Mom: "How's the semester going?" Me: Correct Answer: 21; Your Answer: 19

@dr.rahulgupta7573 2 жыл бұрын

Excellent !! Thanks for presenting a non conventional method for this function .

@PapaFlammy69 2 жыл бұрын

@youngmathematician9154 2 жыл бұрын

If you want a good laugh, pause the video halfway between 4:42 and 4:43.

@desertrainfrog1691 2 жыл бұрын

This is how I solved this integral my first time ever doing it. It seemed the most obvious way, to resurrect the original. Didn't need to look anything up, just messed around with it for a little bit.

@gunmaysingla785 2 жыл бұрын

we can also take one of the functions to be 1 and the other sin^2(x)

@arvindiyer1649 2 жыл бұрын

But you would just be stuck in a loop rip

@desertrainfrog1691 2 жыл бұрын

I don't see how that would make things any better

@gunmaysingla785 2 жыл бұрын

@@desertrainfrog1691 yeah i get it now🤣

@sisyphus645 2 жыл бұрын

Flammy Potter and the Order of the Phoenix 🤧😩

@michaelempeigne3519 2 жыл бұрын

but why not use the formula for cos 2x ? int [ sin^2 x dx ] = - sin x cos x + int [ cos^2 x dx ] int [ (sin^2 x - cos^2 x ) dx ] = - sin x cos x int [ - cos 2x dx ] = - sin x cos x int [ cos 2x dx ] = sin x cos x

@orenfivel6247 2 жыл бұрын

Overkill approach with IBP can be done by differentiate sin^2(x), integrate 1 and use double angle for sin. More over kill is to do u-sub,x=u/2, before or after IBP

@tomkerruish2982 2 жыл бұрын

I'd say that evaluating the gamma function (for positive integers, of course!) is a good introductory example for using integration by parts, since while it's possible to find an antiderivative for the integrand, it's rather tedious (other than n=1 and perhaps n=2). If you wish to stick with trig functions, I'd recommend something more complicated than sin²x, just so it's not something that could be found by inspection or a little experimentation. Maybe tanx sec²x? Come to think of it, you could show that both u=tanx and u=secx work. Just my zwei pfennig! Edit: I should say that you make very good videos, and that it's very easy to criticize from this side of the screen, because both those things are true.

@PapaFlammy69 2 жыл бұрын

@txikitofandango 2 жыл бұрын

My view from the USA is that cos²(x) = (1 + cos(2x))/2 is the easy way to go

@PixelSergey 2 жыл бұрын

I spose the trig identity method is "better" since it can be expanded to higher powers of sine and cosine. Is it e.g. possible to integrate sin²x⋅cos²x using IBP easily?

@stephendonovan9084 2 жыл бұрын

You can integrate any even power of sine or cosine by repeatedly using an extension of this method, and this integral can be expressed in terms of those, so yes

@charliearcaro208 2 жыл бұрын

Great video and such an elegant way of solving this integral, BTW your t-shirt is wrong as the equation = -2 and not 0. I assume it should be e^(i*pi) + 1 = 0.

@siahts2sena 2 жыл бұрын

In Brasil we also integrate by parts! thank you for the video

@gamingstars8956 2 жыл бұрын

Nice i have never solved the integral this way

@_..--- 2 жыл бұрын

how do we get the integral of sin²(x) using the integration by whole method?

@nikhilnagaria2672 2 жыл бұрын

He's got a point. (Also sorry if it's she, read accordingly)

@neptunian6226 2 жыл бұрын

HORSHOE MATHEMATICS for extreme integration!!!

@nikhilnagaria2672 2 жыл бұрын

@@neptunian6226 what

@neptunian6226 2 жыл бұрын

@@nikhilnagaria2672 it's from a funny video called "An evening in the life of a typical math PhD student"

@garvs3thi 2 жыл бұрын

Well we know that (1-cos2x)/2 = sin²x and the function on left is now easy to Integration

@akirakato1293 2 жыл бұрын

There's also reduction formula for integral of sin^n and cos^n which can be proven using integration by part.

@1234larry1 2 ай бұрын

I agree. Using reduction formulas is not learning math; it’s just plugging in numbers and it’s not even memorizing, because you look it up in a book.

@tszhanglau5747 2 жыл бұрын

When you use Phoenix integration: giorno's theme intensifies

@houseflyer4014 2 жыл бұрын

This method is SOOOO much better.

@PapaFlammy69 2 жыл бұрын

yeye

@Nylspider 2 жыл бұрын

Yo Jens, don't feel too bad about the comments from last video. The video was still great and really interesting, just as this one is!

@shadow-ht5gk 2 жыл бұрын

this feels like magic wtf

@pbroks13 2 жыл бұрын

If you solve int(sin(ax)sin(x))dx using ibp twice, then find the limit at a goes to 1, you can solve the integral without having to use pyth identity too.

@pbroks13 2 жыл бұрын

Never mind, you’d have to use the pyth identity when simplifying the limit.

@jorgelenny47 2 жыл бұрын

The Phoenix Integration has no right to sound that cool

@PapaFlammy69 2 жыл бұрын

@ildargainullin7863 2 жыл бұрын

can you use morgenshtern method to solve the integral?

@axels2646 2 жыл бұрын

nice

@darthen8478 2 жыл бұрын

x^3/3 +C

@bhargav7215 2 жыл бұрын

Never disappoints us 🔥

@BrandonUlloa198 2 жыл бұрын

lmao that smooth sponsor transition at 5:44

@PapaFlammy69 2 жыл бұрын

@zildijannorbs5889 2 жыл бұрын

Double angle formula deal is how I did it the first time, I think it’s cool if you’re fascinated by the concept of being able to represent a trigonometric expression in so many ways and when you actually willfully memorize some. It helped to solve the integral, but didn’t help to learn to integrate. I’m all down to learn integrations by parts because t r i v i a l IBP integrals aren’t enough

@IshaaqNewton 2 жыл бұрын

Waiting for Papa to make a Math book recommendation video :)

@tomkerruish2982 2 жыл бұрын

Didn't he do several about a manga math series? Edit: just one about math kzfaq.info/get/bejne/iMljfZupz5zVmoE.html He also did a video about their physics book.

@someperson9052 2 жыл бұрын

I really like that you shit on the Indian education system. There's too much revering of it just because it's extremely difficult.

@johnbiluke8406 2 жыл бұрын

I've never seen IBP done like this...

@luggepytt 2 жыл бұрын

You get 69 points for pathos! Never let someone else tell you what is the “best” way of doing things!

@luggepytt 2 жыл бұрын

But now all the frogs in my garden pond are FURIOUS with you. Their favourite method is to write sin(x) = (e^(ix) - e^(-ix)) / (2i) then square and integrate that boi. They say that you have now insulted them TWICE, and demand a third video, featuring their method.

@youkaihenge5892 2 жыл бұрын

Do sin(ax)sin(bx)dx when a≠b papa! When x on [-L,L]

@glory6998 2 жыл бұрын

I don't know why i try to avoid byparts I always solve these kinda integral by applying trigno identities

@user-tn4qe9ne9j 2 жыл бұрын

Another triviality

@ahmadhaziq3358 2 жыл бұрын

Hello

@veralgupta8182 2 жыл бұрын

I don't like integral by parts much because most questions I have done the equation becomes more complex 😅😅😅

@GearsScrewlose 2 жыл бұрын

Integration By Pork chops. :)

@haiguizeify 2 жыл бұрын

Small criticism: it's hard for me to follow at times because you speak so quickly. I find that I have to rewatch what you've said, but overall this is a good video.

@deli5777 2 жыл бұрын

haha can't please us Americans. jk jk Great vid! My instructor is really good but integration by parts with trig with powers has been kicking my butt

@dadecountyboos 2 жыл бұрын

irly

@syedinayat3548 2 жыл бұрын

Good morning fellow woodworke..... I mean nathematicians wrong channel, Why didnt you set a mimi in the start on your wood working channel We all signed a petition there remember Wood I mean mathematician