Integration by completing the square | MIT 18.01SC Single Variable Calculus, Fall 2010

Integration by completing the square
Instructor: Christine Breiner
View the complete course: ocw.mit.edu/18-01SCF10
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu

[@jasonjackson4555]

The moment she said “trig substitution” it triggered some PTSD from my days as an undergrad. Now I’ll never get to sleep.

[@jeremiaharyeetey9877]

That's sad!!!

[@ShuKatashSam]

I STILL to this day, get nightmares that I did not finish my homework and got behind and have a blank stare at the midterm exam. I wake up and realized I passed and did well, but the PTSD is REAL.

[@loaykher9405]

man, you people act like someone did ask take care

[@tshepisomhlanga5389]

👍

[@oblivion.4974]

Bruh I managed to master trig sub and I was only a hs student

[@saddepressedlemonzest7862]

Love how she's clarifying each and every step and does not automatically assume that we know everything ❤❤

[@is626]

I wish my lecturer was like that.... But he's Indian.

[@saddepressedlemonzest7862]

@@is626Atleast you have a professor. I have to learn it by myself

[@Mahalakshmi-Khan]

@@is626 why don't Indians explain steps?

[@wererer]

@@Mahalakshmi-Khan from my experience ,lot of foreign professors have usually worked with students in their home countries with much higher standards. I had a Chinese professor who just blasted through a course and also realized the mistake he made since back in China, the students that hes generally worked with all were on par and understood the curriculum.

[@LargestClassifieds]

I concur with that

[@Ravi-ot6xj]

I am 32 year old now and I was 18 when I learned calculus. By watching this, my entire teen years flashed in front of me. Thank you.

[@TotalImmort7l]

You still remember the integral tables, don't you?

[@imeprezime1285]

A malo si bacio i drkicu, ha?

[@ThePenthes]

same ! lets watch an another one XD

[@jamesbedukodjograham5508]

I never got the chance to master Calculus back in the year 2000.

[@jamesbedukodjograham5508]

I am totally amazed by the Math situation High School ought to have saved my situation.

[@TheJonathanc82]

The older I get as an adult, the more math fascinates me. I enjoy learning now more than I ever did in high school or college.

[@ligregni]

it has lots to do with being taught correctly...

[@loaykher9405]

man, no body did ask, so shut the fuck up.thank you much love

[@loaykher9405]

@@ligregni learn to teach yourself, loser

[@javierreyes786]

every year that goes by i simply cannot believe i passed these classes only 2-3 years ago

[@Words-of-encouragement.-.]

@@loaykher9405 No one asked for your input either. Maybe you should find someone with a brain and ask them how to type in the English language. Your complete lack of grammar matches your lack of intelligence. Much love.

[@Glockmog2007]

we pay thousands of dollars for classes and then end up having to watch a free video on youtube to learn how to do something properly

[@OM-wb3zi]

@Arsene1412. Oh, I know lots of people that cannot solve a first grade equation and they are employed. I also have seen some others that write comments in KZbin and cannot spell the word -employ- correctly either. I am sure you got a job, right? What you mention then, is just a fallacy.

[@gtas321]

@Arsene1412 You kidding me? I know engineers that's can't even do an integral or a linear transformation. It depends on the specific requirements for each jobs. A fucking systems administrator will NEVER have to use multivariable calculus.

[@patrickthepure]

@Arsene1412 You know what, it doesn't work like that. You learn how to analytically do some sums in college, and then you forget them. In reality, those analytically solvable sums almost never come up. It would be all the same if you haven't learned how to do sums in the college.

[@dadinggo]

I loled pretty hard at this

[@Morgow1]

@Arsene1412 Or write with proper grammar. :)

[@davidahrens2841]

I, as a math major (1970-1974), enjoyed this video very much! After all these years, I'm reviewing my old college notes and using Geogebra to document them. She, as her other video's attest, is an excellent tutor and teacher. I do wish she would have finished this problem to the very end tho...

[@alfiewho1604]

Bravo, almost 40 years out of college, now I finally understand how to do quadratic integration!

[@loaykher9405]

man

[@gabrielroy-manningham5734]

💀

[@bharath8607]

Old man soon gonna be rip 😭 and he's is happy that he learnt quadratic integration💀

[@AC-tn4it]

@@bharath8607 stop hating bitch

[@helin5612]

@@bharath8607 60s ain’t that old 💀 you should be more respectful to older people if you want to be threaten right if you are someday a old man yourself. And you will always learn new things no matter how old you get

[@billk8579]

This young woman is a fantastic teacher. Only a teacher capable of teaching can make something this complex so understandable. I would love to take a math class taught by her. My guess is that she is a graduate teaching assistant explaining what many professors can’t.

[@raps7287]

Bravo lady. Why couldn't my lecturer made it this clear. This lady deserves a cookie.

[@TheChosenOne66501]

*Brava. Bravo applies to one male, Bravi is multiple males (with the possible inclusion of one or more females), Brave is multiple females, and Brava is for a single female.

[@miguelmendoza3403]

JoeQuinn31 brave ¿with the possible inclusion of one or more males?

[@TheChosenOne66501]

Miguel Mendoza Nope. Brave is for a group of girls only. When a male enters, it becomes bravi

[@miguelmendoza3403]

JoeQuinn31 ok thanks for the info

[@swagnikdhar6010]

Tremendous explanation mam

[@bilbobaggins4219]

1/[u^2 - a^2] can also be decomposed as 1/2a . [ 1/(u-a) - 1/(u+a) ] and then integrated without trigonometric substitution.

[@roswelcodiep.bernardo7288]

Yup, integral calculus is full of art though. You can use a lot of techniques.

[@MikeB3542]

This is a recitation session for a first-semester calculus class...so the purpose right here is to apply the principles of trig substitution...force the students out of their comfort zone. I agree that partial fraction decomp gets you to the answer a lot faster, but guessing the students haven't gotten there yet.

[@drgothmania]

If you're smiling happily at the end of a lecture video, you know the instructor is really good. Still enjoy this lecture after watching it for many times.

[@aashcharyagorakh2459]

I can't believe they are teaching this in mit this level of integration is taught to 16 year olds in India

[@innocent718bk]

KZbin recommended this video I have no idea what is going on here or what she's talking about I'm high but its soo interesting

[@securityservices890]

Because it’s rubbish lol all this is just a made up theory to try and prove made up facts

[@joeljoseph26]

@@securityservices890 you are the first human to say mathematics is rubbish.

[@jovidutful]

@@joeljoseph26 w/o math, he'll never be able to use youtbe. hahaha

[@jovidutful]

@@securityservices890 Lol, yeah KZbin is made up of math. w/o the function of it it'll never work, dumbass. just slap your face.

[@taylorlorenztransormation3102]

Welcome to my life.

[@az-nz1pn]

40 years later I can say it still warms my heart to see Integration by completing the square solution being done.

[@CourtneyW-jr6fx]

Integration of the square ...welcome to the "mall" DC

[@nonenone2565]

WOW, I've never seen a college professor who clarifies even the simple steps. This is how it should be done.

[@skyboy123454321]

Many times the professors just complicate things meant to be simple...

[@mawavoy]

I think she is a teaching assistant. Provably a Phd candidate.

[@madhavsharma7222]

This is part of class 12 maths in india. You made it a litle complicated . After completing the whole square formula could have been used directly 1/2a log| x-a| / |x+a|

[@zacharycat603]

if Indians are so smart why are they 112th in the world in per capita income?

[@madhavsharma7222]

@@zacharycat603 Because we have a population of 1.4 billion people. Per capita income depends on the denominator of population. Even countries like bangladesh are ahead of india in this becuase there population is too less as comapred to india. Population of bangladesh is 8 crores less than UP

[@minhaj8520]

@@zacharycat603 memorizing a lot of formulas isn't being smart. Indian students in class 12th had to memorize a lot of integration, and trigonometric formulas, and most of them don't even know how they got these in the first place.

[@Wereld03]

@@minhaj8520 to be fair, i think this goes for math in many countries. Too focussed on learning random formulas without any idea on what they actually do

[@Bhuvan_MS]

​@@minhaj8520If you are a part of a good institution, they will derive those formulae first and then use it as a result to save time while solving the problems.

[@God.Almighty]

so nostalgic watching this. it's been decades but used to be really good at the stuff. she's great, my faint memory could follow her steps for the most part.

[@adabujiki]

SHE IS JUST TOO GOOD... she has so many moves for any roadblock

[@falcongrammarschoolkhanpur9204]

definatly

[@rauhertz2755]

Thats what i call a bad writting and very ugly numbers, she needs more practice at the blackboard.

[@arlenestanton9955]

What !?Her numbers are great. I’m not an expert on writing on a blackboard, who is. I found the number readable

[@atgblue1]

I thought this was going to be a math video. Turns out it's a foreign language class

[@mrstatler]

"Sorry, I don't understand Welsh" "This is English" From some movie.

[@rollotomassi7437]

@@mrstatler More like Greek

[@loaykher9405]

you're funny

[@chuckwilliams8738]

She is am amazing teacher. It's no wonder the best minds come out of MIT. They have the best professors. I wish that my instructors taught this well.

[@MrTrollo2]

I think people underestimate how much easier it is to "understand" a concept you already know. Of course every of her words makes painfully much sense when you know how completing the quare works. But you can never evaluate again how much you would've understood if it was a new thing for you.

[@raghavkumarsingh4222]

I dont know the foreign curricula, but we are trained more on calculus at +2 level -(17-18 years age). Please refer JEE ADV problems of calculus, then u will know what the point is!!!

[@toby2668]

I finished Calc 1 through 3, yet still came back to watch these videos... Just so fascinating!

[@injamhimu2719]

"So this is some constant that we don't care about..." Epic one🤣

[@philiptownsend3622]

well, it's not hurting anyone...

[@williamogilvie6909]

Great recitation! Very clear explanation and easy to understand. This instructor is gifted.

[@UrielX1212]

Good math teachers are few and far far far between. This is an example of a stellar one.

[@TERRYBIGGENDEN]

I wish I had her for maths at uni. I had to watch this three times. but I followed it. I still need to try and compete it (many decades since I did this kind of thing). Shel's good-knows what she is doing and has personality. She reminds us of important steps and why they are there and is encouraging. A great tutor. :-)

[@genki2705]

my professor: you already know that ... no, we won't do that, that is too easy ... trivial ... and that is also trivial ... two hours later ... we've finished half the book I still have nightmares

[@deewilliams4000]

How she wound back to the start(including bit of side talk), then did it step by step....marvellous!

[@cogitoergosum7945]

MIT know how to hire the best professors. She is very good at communicating her thoughts. I am working it hard right now to get admitted for masters degree at MIT in 2023.

[@altbust]

Been 25 years since i did this stuff in college..haven't touched calculus since.... but watching this video has reminded me what a pleasure learning can be when you have awesome instructors like Christine.

[@swadhanandy9666]

Yes solving integration problems are a source of pleasure

[@swadhanandy9666]

Genuine pleasure

[@swadhanandy9666]

Genuine pleasure

[@tahisev5756]

If I could watch these videos in my college days, I would do better in some courses. She is an amazing teacher. ❤️

[@crimony3054]

So when you see this problem on the test, save your time and go to the next question.

[@markh.6687]

The next question: "Calculate infinity." Me: FFFFFFFUUUUUUU!!!

[@oxfordmathematics6835]

kzbin.info/www/bejne/f4GsoJ55Z91pn5I

[@oxfordmathematics6835]

kzbin.info/www/bejne/f4GsoJ55Z91pn5I

[@Kneecap22]

Yeah, I solve it using Wolfram Alpha. And even if you know the input it still take Wolfram 20 seconds to load. Input: D[-(Log[(-4 + Sqrt[15] + x)/Sqrt[1 - 8 x + x^2]]/Sqrt[15]), x]

[@nicholasschroeder3678]

I was going to make some smug stupid remark about how I was so glad I didn't have to do this stuff anymore, and then I got sucked in, and a flood of nostalgia came over me. She's a wonderful teacher.

[@motonationarmy9083]

Such an excellent teacher! I’m taking trig rn for my physics major and I’m able to follow this so easily!!!

[@samb2166]

wow. it's been 10 years since I took calculus, and her clarification brings me back memories!

[@leecoates]

By far the best explanation of this on the internet that I've came across.

[@kumarupendra]

kzbin.info/www/bejne/r5C7qIShmM2qqKc

[@dneary]

I never remembered the trig sub for 1/(u^2-a^2) - I've always found it easier to use partial fraction decomposition 1/(u^2-a^2) = A/(u-a)+B/(u+a)

[@sujitsingh7279]

I used to do the same when I was in school.

[@stelioskatsampadimas1693]

YES MY DEAR. SIMPLE FRACTIONS DECOMPOSITION, AND NATURAL LOGARITHMS! THE LESSON IS NO GOOD :)

[@valuedhumanoid6574]

Yeah, me too. I was going to say that exact same thing. And I was going to work in Pi and Ackermans function with Planks Constant to make it sound even more thoughtful and smart

[@charleskomo5306]

En effet il est aisé de passer par la décomposition en éléments simples comme vous le suggérez.

[@roshanantony7467]

SAME!!!! I used to do it this way!

[@toomanyhobbies2011]

Love the chalkboard. We were using markers in the 2000s and now they're all using digital drawing boards. She's a TA working off her tuition and having a great time doing it! For those of you learning from this, just remember that Calculus is Algebra with a couple of new concepts. If you know your Algebra, Calculus is fairly easy. If you're studying Physics, you get to move into Calculus of Variations, Complex Analysis and whatever else helps you solve your advanced problems. ;-)

[@sid_1724]

I'm not math student but based on my little school knowledge I understand this problem with great explanation....step by step. ❤️For professor. It was magic ✨

[@NNCCCC63]

great teacher ,.... not everyone who has knowledge is able to share it effectively and offer lucid explanations.

[@acershund1]

What a great instructor! Enjoyable to watch and learn from her. A happy professor with a great smile and great knowledge.- How can you go wrong? I would always pay close attention in class with her at the blackboard!

[@jamesbedukodjograham5508]

She is a great teacher and we appreciate her achievement like we do in Ghana especially her stance on the Calculus numbre Theory.

[@williamquintero981]

Excellent analysis.Always a good thing to remember.

[@cooldown369]

I am a working professional now but I was compelled to solve this problem. ❤😘Studies never go in vain though not in use.

[@slackerlifeman]

This popped up in my suggestions and I'm sitting here wondering how I ever passed.

[@abdullahimohamed4006]

You made a difficult issue very simple. Thank you so much for the great job. I just would like to point out a small piece of problem: when you are explaining the numbers, a subtitle covers them up and I find it worth mentioning it, so that you can take it into consideration for the next lecture.

[@the_biggest_chungus7508]

So I'm a german student, studying electrical engineering. I'm not the most skilled mathematician, but curios and love the way how almost every science problem can be formulated as a mathematical equation. And ma'am, the way you teach is amazing. Explanining everything friendly and step by step, while this hyping "common, that's easy, I'll show you" kind of vibe catches you. Interesting how a maths lecture of the worlds most prestige College is easier to understand than the same lecture in a german upper level college

[@zonda456]

2 am. Having beers. Drunk. Watching this lady do math

[@VLSrinivas]

my teacher asked us to remember the formulae and substitute in a single step. We never wrote such detailed expression anywhere. Its really fun to see her elucidating step by step.

[@oxfordmathematics6835]

kzbin.info/www/bejne/f4GsoJ55Z91pn5I

[@mersenne2486]

yes that 1/2a(ln|x-a...... formula

[@prody666]

Knowledge should be available to anyone, not only to those who can pay. Thank you for sharing it freely.

[@MasterCivilEngineering]

Great see here more

[@mark0032]

It is available to everyone, universities don't own knowledge, you can buy all the books they have.

[@johntrapp5344]

This was an excellent video. I’m sure it’s just me but I don’t remember trig substitution being this seamless (not easy…seamless). Learned a lot here

[@mr.jaydavidson1225]

I love this teacher. She's amazing!

[@giuseppepippoizzo6506]

Nice explanation, I never went to high school, but found your teaching easy to learn.

[@brainsterind]

Lovely problem incorporating a number of techniques to solve...great refresher for those of us that have been away from calculus for a little while! 😊🙏👍

[@regdor8187]

A "refresher" implies it is something of little use, easily forgotten....

[@RegisteredLate123]

This is like absolute basic integration buddy I bet ur american

[@cwldoc4958]

Excellent video, very clearly explained. That having been said, this particular problem can be solved on one side of a small napkin by rewriting the integrand as [1/(a-b)] [1/(x-a) - 1/(x-b)], which immediately integrates to [1/(a-b)] ln|(x-a)/(x-b)|, where a=4+sqrt(15) and b=4-sqrt(15).

[@tristacker]

I'm relieved that I don't have to remember this kind of thing for real anymore.

[@HashmiAcademyOnline]

She's great. clears everything up for me. :)

[@arefinazam4341]

double meaning lmao

[@user-sy3zv3ex9v]

You should use 1/(x^2 -a^2) = 1/{(x-a)(x+a)} = [1/(x-a) - 1/(x+a)]/(2a) . The you can easily get [ln(x-a) - ln(x+a)]/2a + C, where C is a constant.

[@margaretm.7079]

Excellent presentation! Thank you.

[@mayukimura1894]

Ohhhh, actually I am taking PreCalculus and I don’t know about these stuff, but from my knowledge of trig and wonderful her explanation, every step really makes sense for me :)

[@kxmode]

Great thing about math is this will still be valid in 50 years, even if Google and KZbin cease to be.

[@SpaceboyGT]

im proud to have made it thru this video. its been like 12 years since and passed with a D. ima a total dyslexic with numbers on paper and the language sin, cos, tan, all foreign.

[@markh.6687]

Should have seem me try taking a high school Physics class not even knowing what sin,cos,tan even were(!). So much for calculating force vectors and such. I understood the theory, but had no hope without enough math classes to do the assignments. I did one problem with a fraction under a radical sign backwards once because that was the only way I could use my weak Basic Algebra skills to even deal with it; took 5 hours to do it. Teacher later re-did the work I turned in, and very politely concluded that somehow all my mistakes canceled out and I got the answer right(!) Teacher was great, once he knew how hopeless my math skills were, he went to Guidance, and told them to move me into Study Hall, because all I was doing is failing, and asked them why they ever let me sign up for Physics anyway?! There are some odd things I can do, like take 10% of a number in my head, but for the love of Sir Charles Babbage, don't ask me to show my work, because I can't do it. I also got rather good at geometry proofs in Basic Geometry for some odd reason.

[@jadrianlane]

I admire your ability to disclose your ineffectiveness, I really have to struggle to disclose that imperfection.

[@oxfordmathematics6835]

kzbin.info/www/bejne/f4GsoJ55Z91pn5I

[@electrolockinc]

Can't get enough of the simple explanation to explain & have some fun with daughters, TY!!!

[@ROX_AN_100]

Wow she illustrates the examples in a wonderful and beautiful way♠️🖤🖤♠️

[@edschaeffer]

If I was doing this problem I wouldn't have used trig sub. It may have been easier after you made the u-substitution to just simply use partial fractions after the u-substitution. From 1/[(x-4)^2 - 15] : u = (x-4), thus 1/(u^2 - 15) = Integral 1/[2*sqrt(15)] * { [1/[u - sqrt(15)] - 1/[u + sqrt(15)] } Ans: 1/[2*sqrt(15)] * [ ln|u - sqrt(15)| - ln|u + sqrt(15)|]+C ... Student can substitute.

[@vikku_19]

edschaeffer yeah it is better.

[@ADendane]

Yep much faster

[@mauriceoreilly9955]

It's really misleading to use a trig sub! Misleading also to say that x^2-8x+1 can't be factorized! x^2-8x+1 = (x-a)(x-b) where a = 4+sqrt(15) & b = 4-sqrt(15). Then anti-derivative is (1/k).ln[(x-a)/(x-b)] where k = a-b = 2sqrt(15). Much simpler ... and no trig! Moreover, checking the result by differentiation is a nightmare using Prof. Breiner's method. Just try it! Differentiating ln[(x-a)/(x-b)] gives (by chain & quotient rules) (a-b)/[(x-a)(x-b)] and the checking is done. Regretfully, the method shown in the video demonstrates 'hammering' rather than 'understanding'.

[@jayantjha3128]

Tbh she was trying to teach a new method and wanted students to try by that method

[@gybx4094]

Imagine if all of our politicians were required to be this intelligent, honest and congenial.

[@Galbex21]

Math politicians would not be able to understand social or historical problems. Intelligence in math is not related with general intelligence. They would just be politicians that know how to do math but they would probably be incompetent in other fields.

[@erikbakker1531]

That would be a better world.

[@samarth6817]

Wow!!! I wish I was explained Integration the way you just did. Thank you miss. You are amazing.

[@laravieira2117]

incredibly easy with her explaining. ❤ IT'S BEAUTIFUL WITH ALL THIS RELATIONSHIPS. Love math. Love this teacher

[@hashimelti7355]

I am in love Kristen, wish my tutors were as talented as you're... kudos

[@jadrianlane]

I agree what a wonderful woman. I hope there are lots more like her. She an National Treasure.

[@vychuck]

Christine thanks for helping students, it's great

[@user-tj7rt1tg6z]

perfect, congratulation, 80 years old math. prof, 40 years in Czech, 40 in CanADA

[@freddycarias6464]

Thanks so much! You are a great teacher. It's so clear.

[@factChecker01]

This is a great example of how KZbin allows us to learn from the best teachers.

[@cwldoc4958]

No, this is an example of how MIT allows us to learn from a great teacher. KZbin is just a middleman who charges for it (through ads)!

[@factChecker01]

@@cwldoc4958 , Good point. But it would not be practical without the internet and something like KZbin.

[@YemYum]

just finished calc 3 in college. Im so happy i dont have to think about this shit anymore.

[@celiogonsalves6758]

Ótmo video Professora, está de Parabéns.

[@jerryadriano8837]

muito bom assistir essa aulas, pois melhora e muito o listering no ingles

[@babakpakbaz]

40 yrs ago I had math teacher. who used to give us 20 home work a week, solutions was posted on Monday morning. in 3 semester I did more math than all my family done for 100 years. make story short went to get two masters degree. and one heck of life, I rem. him for ever.

[@pepecohetes492]

Very didactic, well explained, excellent professor. Thanks.

[@xtiago16]

Very nice , I´m from Brazil . liked class . You are a great teacher ...

[@AndiAli]

I enjoyed this. Beautifully explained.

[@emproduction4619]

Excellent teacher with excellent methods !

[@Lucerozeus]

Oh my... awesome professor. Great lecture.

[@huh4157]

Man this is the complete summary of algebra trig and calculus classes which are combined of 200hr + ☠️

[@rodrigorebolledo2894]

Por eso son las mejores universidades del mundo por que lo resuelvan de miles de maneras y lo hacen aún más complejo para la resolución de un problema real

[@multimathematics]

An informative and comprehensive lecture

[@otsiletaut-man365]

WOOOOOW!!! this lesson is awesome,i love it,you are a geneous!...i was really shocked coz i did not understand what the lecture was diong in class n i tried by myself,still getting nothing then i decided to watch this vedio.that's where i found you n now i understand this better n i can solve many related problems...THANK YOU FOR YOUR ASSISSTANCE!...A REALLY APRETIATE THIS VEDIO....#THANK YOU AGAIN N AGAIN

[@RiceNinja16]

THANK YOU! clears everything up for me :D

[@flickerduck3495]

hey it's been 9 yrs, how r u mate ?

[@crossbonez1028]

@@flickerduck3495 I’m very curious how this persons life is. This person could be anywhere and have a job of anything in the world right now

[@juandiaz8558]

Thank you very much I appreciate a lot your work, keep going

[@wjimenezu]

2:00 Pricesless trick.... thank you so much!

[@chuckszmanda6603]

i'm an old guy who still loves this stuff. Great job!

[@oxfordmathematics6835]

kzbin.info/www/bejne/f4GsoJ55Z91pn5I

[@SuperLeonardo40]

I admire you.I was bad in maths at school.Really bad.God, i didnt like it at all.My teachers also disliked me.I passed all the grades and years.Never had to retake a year at school but it was really heavy for me.That blackboard , full of logaritms, and other stuff in high school.Didnt remember much, after.Today, you ask me something of that and i woud have to study again.Why was i like that?.

[@patrickmetthew7830]

How my father friend

[@bahagharidon1940]

The board space saves me on this, THE THUG LIFE choose the board space on this.

[@imbicyl6667]

please can u tell me your solution?

[@fannypack7705]

@@imbicyl6667 lol

[@norraihanaasmarmohdnoor4024]

Imbicyl 666 ssfggffugghuiihghuhhgffghGphzgaG:));),(>)3667987666()

[@asystole_]

what?

[@tahirnaeemdynamicchemistry5762]

kzbin.info/www/bejne/oYjKp2Sdlp2bi6c

[@AnalogAirwavesWAAIR]

This instructial video will age well. Thank you for sharing

[@KeiLin11]

Students are very lucky these days to have this brilliant educational materials

[@sergiog2930]

One teacher like her is what i needed ar high school

[@mauriciougaz]

That's what we all need dawg

[@gulustanmustafaqizi6816]

This method of solving liked me. I learned a lot

[@georgesadler7830]

In Calculus, this is where each student algebraic skills have to be really good in order to solve these types of problems with mathematical tricks. In order to excel in mathematics , students must know algebra from top to bottom. Professor Breiner demonstration of Integration by completing the square is one of the techniques of integration.

[@jednorazowy1000]

But very easier is also algebraic decomposition: en.wikipedia.org/wiki/Partial_fraction_decomposition#Example_1

[@d.a.6911]

This is wonderful, a sign that there's a lot for me to learn😍

[@Danny-fs1hk]

I’m having college flashbacks dating back 31 years.

[@FLIGHTCOMPANY]

Professors always make their integral sign perfect. Mine looks like a Bambi trying to walk for the first time.

[@bbb_888]

It's literally supposed to be an "S" stretched vertically.

[@matambale]

or a baby giraffe.

[@petrofilmeurope]

Excellent presentation from a pro. Thank you from Oslo.

[@abdelrhmanawad5449]

I like her style in teaching Much respect from Egypt 🇪🇬

[@pacoespinoza4248]

JUST HERE TO IMPROVE MY ENGLISH ABOUT NUMBERS. THANKS A LOT.

[@beoptimistic5853]

kzbin.info/www/bejne/rHenfpR-hpmejZo 💐..

[@messemphanger]

She could come back 20 yrs later and I'd still be trying to figure it out.

[@nubianpwr]

LMAO:-)

[@vicks711]

That is why your middle name is moron.

[@messemphanger]

@@vicks711 That's why you are an asshole.

[@AlessioQ]

very very clever!!!!!! I would like you to finish the problem till the end, but it's ok!!! love how you explain every step!😀

[@s.i.mamunhossain1901]

Excellent! Today I learned calculus in details. Thank you.