How to Integrate Using U-Substitution (NancyPi)

MIT grad shows how to do integration using u-substitution (Calculus). To skip ahead: 1) for a BASIC example where your du gives you exactly the expression you need in order to substitute, skip to time 1:30. 2) For an example where you have to REARRANGE THE DU by multiplying or dividing because the du has a different number or sign than what you need, skip to time 8:21. 3) For one where you have to REARRANGE THE U by subtracting or adding because the du expression cannot give you the expression you need, skip to 17:15. 4) For u-substitution with TRIG (SIN/COS) and the power rule, skip to 22:35. Nancy formerly of MathBFF explains the steps.
Follow Nancy on Instagram: / nancypi
Twitter: / nancypi
For my video on the BASICS of INTEGRATION, jump to: • Basic Integration... H...
With all u-substitution integration problems:
The FIRST STEP is to pick your "u". The best choice is usually the longer x-expression that is inside a power or a square root or the denominator, etc (in an "inside function"). Set u equal to this x-expression.
The SECOND STEP is to find "du" by taking the derivative of the u expression with respect to x. For instance, if you have u=3x+2, your du would then be du=3dx. **NOTE: Remember to include "dx" at the end of your du differential expression.
The THIRD STEP is to substitute u and du into the integral everywhere in place of x and dx. **NOTE: if your du does not perfectly match what you need in order to completely substitute before integrating, you must rearrange the du, or sometimes rearrange the u, in order to fully substitute before integrating. For an example of each, see example #2 (time: 8:21) and example #3 (time: 17:15).
The FOURTH STEP is to integrate, remembering to add "+ C" at the end since you integrated an indefinite integral (no limits). The du goes away when you integrate.
The LAST STEP is to "back-substitute" by replacing everywhere u appears with the x-expression that you chose u to be.
For more of my math videos, check out: nancypi.com

[@melissag.1795]

Does she have anywhere where we can donate to her for providing these wonderful videos? Made this so easy to understand and was way more helpful than my professor. Thanks so much for making such quality videos and going step-by-step. It makes those of us who are taking classes a decade after we finished high school much easier to understand! If you have anywhere I can donate to you for your time/help I would love to know!

[@NancyPi]

Hi Melissa, yes here you go. I'm happy they've helped you! 💛 www.patreon.com/nancypi

[@ericguevara564]

couldn't have said it better my self

[@SWEETG]

Sooo true.....wayyyyyyyy more helpful than my professor 😃

[@klutchboi3266]

@@BST-vk7lb lmfao

[@aaronmckinnon12]

u dirty dawg, we all know the real reason ur donating

[@evilum]

can i substitute u for my professor

[@feero9680]

evil What the heck r u....? I see u in every of her video. Youre learning, eh?

[@evilum]

idk man

[@reynoldmaranan6839]

Nice one

[@reynoldmaranan6839]

What grade are you?

[@Ko0okieeZ]

Cleaver af

[@androidandroid9591]

The best thing about Nancy is that she never, never, goes into full math professor mode and says: "and it is obvious that...". She understands that if you are looking at these videos, there is nothing obvious about math. She does not skip three steps or assumes that you will figure it out. By far, one of the best teachers on this subject on line. Hope she will post more (she probably graduated already and is pulling mid 6 figures at some super corporation).

[@edwardthiccums2000]

I also like that shes super sexy.

[@soumyajoypal9359]

@@edwardthiccums2000 the main reason we come herw

[@user-fk1cs2oh5i]

@@edwardthiccums2000 that’s probably why she doesn’t post anymore.

[@tis_i_sana]

YES this is why the teaching at Oxford is so bad !

[@th3azscorpio]

A lot of math teachers/professors are dicks for some reason. I appreciate her approach.

[@jhilikjhinukb873]

Nancy, not only were you a great source of clear information and support during fairly trying times-the pandemic has pretty much wrecked our schooling system-but you're also a great source of inspiration and clear-headedness for all of thus. Thank you so much for you invaluable content.

[@SpitKid]

i have so much respect for this girl for being able to write everything in this entire video backwards just so we can see it the right way.

[@shlarry3184]

Are you being subtly humorous? I can't tell, but I wondered the same thing. I might be slightly stupid, but I'm pretty sure that's not a thing.

[@SpitKid]

@@shlarry3184 i was joking halfway lol. I thought she rlly did that until i realized she probably wrote it normally but just flipped the video

[@shlarry3184]

@@SpitKid It's an interesting chirality proof brain exercise.

[@WearTech]

@@SpitKid Look at the dry erase marker. EXPO is the most common brand of dry erase marker. The marker has the word "EXPO" like backwards ("OPXE") written on it. So it does look like it was either left-right reversed or it was shot with a camera pointed at a mirror.

[@andrewmontoya8511]

man u a special kind of stupid

[@johnbyrne4599]

Would totally substitute U for my X.

[@realmadridfan9949]

john byrne corny

[@johnbyrne4599]

I couldnt resist, hope she gets a chuckle outta it

[@sudoaj]

Bad joke

[@moebazzi8412]

genius

[@SWEETG]

Smart joke 😂

[@abzshaker]

I feel like she is using hypnosis to make me focus on maths

[@takyrica]

😅😂 is that why I’m watching her videos even though I need the math for nothing? 🤔😂🤦🏾‍♀️🤷🏾‍♀️

[@Cdcenter1234]

kzbin.info/www/bejne/d4aXiXiIeLWdf9U

[@Cdcenter1234]

@@Jacob-ed6wv kzbin.info/www/bejne/d4aXiXiIeLWdf9U

[@elgeroz8732]

So true 😂

[@brrrrr256]

A smart girlfriend is the best way to get through calculus. Or, it was in 1987.

[@bigseekersb]

I have retired form full-time work and want to be a math tutor, to work part-time. I took calculus over 20 years ago and I am relearning it. Nancy, your vids are crystal clear. You are slow and calm, and I love that. Great work! Thanks!

[@maryloelr.sanchez1242]

Out of all the tutorials I've watched? This is one of the most effective ways of teaching and as a student, it helped me the most to understand every detail about the topic that I am looking for. Hoping for the next problem tutorial. Thank you so much.

[@bicboi46]

"Are the videos from the 2006 really helping???, yes KZbin advertisment it is, now leave me alone jesus.

[@dota4lyf224]

*jey sus* math does not change

[@offenseliontv8803]

jimi hendrix nice

[@SDCustoms]

Just try STUDY PUG!

[@avi-brown]

had this exact thought!

[@suly4346]

swear to god if i get that ad again im gonna find the person who made it and shove my 'A' grade test paper in their mouth

[@karlesmcquade2863]

I've studied calculus and I'm a teacher. These videos on calculus are easily the best series on the topic I've ever seen. Nancy relates easily to the real struggles of calculus students, each step in the approach connects clearly to the previous one, and her set-up of the clear board looks awesome but is very effective and non-distracting. Beautiful work.

[@Blalalala-xd5tl]

@BallerJack05 nah she’s using mirrors

[@xhajssoa9565]

Or maybe ur a SIMP

[@absentgraph9864]

@@Blalalala-xd5tl yeah,in the vid she s writing with her left hand,but in reality its ther right hand

[@monicam.worden2916]

Nancy! Fewer tears have been shed after I discovered your videos. I'm beyond grateful.

[@danielcervini2545]

WONDERFUL !! I just went through this entire video very, very slowly, making sure I understood absolutely everything that was written. And it was almost fun. I think this is as painless learning how to integrate using u-substitution will ever get. Thank you. Thank you. Thank you.

[@itsabelleee]

Took Calculus 1 two years ago and Pre-Calculus nine years before that. I'm in Calculus 2 now, and I had no idea there were such things as a "refresher course." Needless to say I've been trying to re-teach myself Trigonometry/Precalculus in order to relearn Calculus 1 in order to even grasp Calculus 2. My professor has been explicitly asked twice over the past few weeks, "How do you know what to take as u?" and his answer was the same each time: "Think of it as... an art. You can't really explain. It's uhhh, intuitive." (What an educator.) Sigh. These past 3 weeks have been rough. Like, really rough. Thank you for softening this hard life♥︎ You (and Professor Leonard) are incredible at translating this crazy conceptual language that is mathematics into sweet plain English. Thank you thank you thank youuu. My first exam is in a couple days and it's a sweet feeling knowing I might actually pass 😭

[@danny89620]

Did you pass

[@itsabelleee]

@@danny89620 Yes I did - actually aced the first exam this topic was in :)

[@juanbarrera3722]

@@itsabelleee how was calc2 ? is it as bad as they make it out to be? i’m approaching the final for calc1 this sem and i’m pretty nervous for calc2 from all the stories i’ve heard

[@solidus8098]

When she explains something complex, suddenly it becomes simple. Teachers around the world need to take this girl as an example. Please keep up the amazing work! You got my support until the end!

[@GiorgioAngeloRuntuwene]

Solid Chrome i

[@kumarupendra]

kzbin.info/www/bejne/r5C7qIShmM2qqKc

[@gunjanrana4474]

Thing is that boys concentrate more so they understand easily.

[@aadyadubey379]

@gunjanrana4474 as a girl I can confirm that I concentrate more than boys do 😏😏😏🏳️‍🌈

[@aadyadubey379]

@gunjanrana and i am pretty sure I am not the only girl to concentrate more than boys....there must be many girls like me... Especially teenagers like me 😏😏😏😜

[@zanedavidson1497]

Usually, I get distracted by the teacher if they look this good, but this channel has the opposite effect on me and helps me stay engaged and listening. I've come to enjoy topics I normally wouldn't have. Thank you so much!!

[@lilig4778]

I love your videos, they're always the MOST helpful out of every single channel on youtube. You're a great teacher thank you!

[@drewpetersen3683]

I really like how you provide the different "types" of problems you will see when using this process. I know you do that with all of your videos, and it's really helpful for us to learn that, even if we are practically copying when we see it in application, but it makes it easy to catch on how it works in general later. In other words, the way you make your videos is probably the most helpful from what else I've seen on KZbin. Thanks!

[@Bhamilton-ws4go]

I'm extremely impressed with your ability to write everything in a mirror image perfectly

[@kenmendoza6932]

Bhamilton0347 . I like how clueless you are.

[@zacriversjr7029]

Silkie Shag It’s flipped along the x-axis. Rotation wouldn’t do..

[@silkieshag9703]

@@zacriversjr7029 you are right , i made a mistake, but i think people understood what i tried to say. thanks for the correction.

[@zacriversjr7029]

@@silkieshag9703 😂 No Problem

[@zeeshanzaffar1435]

You are innocent as a 🐰

[@stanleyngugi1675]

She had me at "You don't have to like math, but you can like my video"..!

[@fionahysa]

I’ve watched so many of your videos to help review and better my understanding for my Ap Calculus Final and your explanations and step by step examples have helped so much, i’ve better understood the concepts in the way you teach than any of my teachers in the past, thank you so much Nancy!!

[@swindler1570]

Nancy, thank you so much for saving my sorry butt again. You got me through calc 2 three years ago, and now you're helping me review for Calc 3, Intro to Linear, and Statistics for ECE. Love your stuff - very clear, very concise, easy to understand. You might just be the only reason I get through this hellscape of a term.

[@ac4190]

"What happens when you have a trig function?" Well, I simply cry. However, this video was beneficial.

[@fallout3freak360]

Hah, if you don’t like trig integrals you won’t like calc 2

[@ac4190]

@@fallout3freak360 You're right, I did not 😣

[@TheEndernal]

lol

[@rimshakhattak7269]

One of the best teacher ever... Such a calm voice, calm way of teaching.. Great method of explaining things..

[@seanevans2700]

Really impressed by your informative videos. You have made something that has been very difficult to understand in the past to something that is much easier to learn. I believe I have the ability but never had the right teacher to to break it down in such simple form. All students have different learning abilities. I think the problem is that some students are embarrassed to speak up. Keep up the good work!

[@peterv1806]

Nancy - you explain everything so well. I wish that I had had a teacher as gifted as you when I was initially learning these concepts forty years ago. Thank you for all that you do. Please keep up your amazing work.

[@HouseTre007]

the detailed ques you give are what someone who took these courses many years ago needed for it to all come back, ty

[@Killernochance]

You are an excellent teacher, especially for people who are new or hardly know where they are going. Thank you!

[@atheistsanonymous5376]

Blink twice if they're holding you hostage.

You noticed she seemed upset halfway through, too, huh? Lmao

[@hunter19884]

Spray some pepper if they are causing you lust. - The Holy Fuck1ng Bible.

[@ValAi178]

@ I am sure writing mirrored on a glass is not satisfying.

[@navjotsingh2251]

They are holding her hostage, on the other side they have a gun to her and the only way she can stay alive is by doing calculus.

[@jaredburgon8155]

@@ValAi178 she actually writes normal to her perspective, then they flip the video when editing. you can tell because her expo marker letters are mirrored

[@ZippyCoyote4]

Thanks to you and your old channel, I passed Calculus 2 so I don't really need to watch this video anymore. But I still wanted to give you the recognition, views, and likes that you deserve! Please keep the videos up. You are helping so many of us struggling students!

[@brookestephen]

you make reading & writing backwards look easy! Nice to finally see codified WHEN to use them - Thanks.

[@lanomusambazi8654]

I love your teaching approach it's really great. You really introduce the concept nicely.

[@nathanlee3077]

My lecturer spent an hour trying to teach this and just made me more confused. This is excellently taught and I now feel far more confident in using integration by substitution. Thank you so much.

[@russianswag409]

Nancy, you're such a wonderful teacher! You have really simplified the concept of u substitution for me! So much clearer now. Thank you so so much! Way better than my lecturer at university

[@allanfraker5701]

She made this so much easier. I know my college professor has to get through all this in like 40 minutes and show multiple examples so I understand its hard for him to go through step by step like this. So to the people who take their time to go out of the way and post videos to help all of us who struggle, Thank you so much.l

[@ryanmacewen511]

WOW! Perfect presentation. I love the slow tempo. I've been out of school for 8 years and needed a refresher for going into Calc II. This was perfect. Thank you so much! LOL I thought I was good at writing upside down, but you are awesome at backwards. Very clever low tech approach. cool!!

[@jackrotter9434]

Nancy you are awesome at teaching! Especially during this pandemic and with online school! I hope you have calc 2 videos so I can watch those next semester!

[@mrch33sehed93]

That was seriously satisfying! A phrase i never thought I'd utter 🙂👍

[@anthonyali3838]

Nancy, I am quite impressed with the way you teach this topic. Please, keep up the good work!

[@ashimthapa.5524]

You made a great impact in my studies, I started derivation and integration with your video and you made it so easy. I am very very thankful the most beautiful teacher.

[@captaintimcurry1713]

You're never allowed to go away, Nancy! I swear, I can struggle with a calculus concept for hours, but watch just one of your videos and suddenly everything is crystal clear! You make it so easy and I love your teaching style! Thank you!

[@dadanramdani4425]

were she my lecturer, I'll never skip a single class.

[@asianboyy3839]

Get your grammar correct first you simp

[@good-tn9sr]

shouldn't even be skipping classes smh

[@alpharibo]

I just refreshed my entire calc 1 class by watching these videos and I understand it way better. This channel literally is saving my college GPA

[@OGDEN25]

I like how u make us feel like as a student you didn't understand certain steps and explain it well.... Love you♥️♥️

[@SunriseLAW]

NancyPi is my girlfriend and she is just like the square root of negative 100: a perfect 10 and imaginary.

[@parthmaheshwari3323]

No other comment had been more math-ish.

[@ankitaaarya]

@@parthmaheshwari3323 parth Gaming

[@parthmaheshwari3323]

@@ankitaaarya Yes?

[@reversev9778]

Parth Gaming parth Gaming

[@dominator29lucky]

Clean shot

[@johnbowman9734]

Thank you so much for taking the time to make this video!!!!!

[@rosellaaj3709]

Nancy pi thank you so much for your help and hard work and sweetness and just having a passion for helping people who are struggling with math 💛💛💛🌟🌟🌟 You saved my understanding for over a year now

[@quintonwilson8565]

Your attitude/approach here is very smart and reassuring. Great calculus lesson and great professor.

[@tshegofatsomatee]

Honestly thought u-substitution was a difficult and I was quite intimidated by it but you explained it so well and I now have a better understanding of it🥺🥺thank you for this

[@blackshout8841]

it is easy just need practice to be perfect

[@MicrophoneLion]

Every math KZbin out there is the nicest person i know

[@xana9849]

ive never been so intrigued by the u-substitution way of of math! It is so compelling and her boo-manerisms are very great at explaining the topic.

[@balbirrao3551]

Thank you Nancy for this. My daughter is having a hard time at school with this, I have learnt now, will teach her and ask her to refer your videos. Gratitude!

[@neural8885]

No jokes from now on. IT’S STUDY TIME

[@mitulbhale7519]

Bhai kuch nahi idhar se mat padh ullu banari wo looks dikhahkar integration padhna hai to umacademy se padh

[@trunkten5452]

are bhai bhai

[@juanpedraw4245]

HAHAHAHHAHAHAHAHAHAHAHAHAHHAHAHAHAHAHAHAHAHHAHAHAHAHHHHAHAHAHAHAH

[@TheRealDunks]

it took 8 minutes to understand 3 classes of Calculus 2. My professor needs to see this amazing educator!

[@abhigyanshukla3233]

Thankyou Nancy Ma'am. It were you because of whom i was able to understand integration by parts as i was looking for this topic to master for physics. Thankyou once again Ms. Nancy!

[@stevenrichards1539]

I never thought I would fall in love with math and then NancyPi.....

[@cayezara8110]

Nancy, When you explain, it is clear as crystal. You are beautiful!

[@tanguyrazafindragolo6628]

I’m paying $2k for my Calc 2 class. Attended 2 class of 4 hours and thought I would drop the class. Now I’m watching this video and I feel like I can tackle calc 2! Thank you, you’ve helped me since pre-calc

[@benjaminsibson4265]

Why’d you pay 2k bro :(

[@or1343]

2k??? where is that

[@AK47_414]

@@or1343 university/colleege classes in the United states

[@514musique]

2k is a steal compared to other college prices

[@Mike-xu2qh]

Thanks for this. You explain 50 pages of convoluted textbook gibberish in 25 mins. I'd fallen behind and really needed a refresher. Really grateful.

[@jelinayengkhom8478]

Thank you so much for making this video,, i couldn't understand something since the last 2-3 days because I don't have a good basic foundation,,, it is so helpful to me ,,, completely cleared my doubts❤️

[@laman__samo]

Honestly GOD BLESS YOUR SOUL😭❤️❤️❤️

[@mms7146]

5:45 that's where it just clicked for me! thank you!

[@hildahchanda4827]

Wow this video has been so helpful. The way you explain step by step so calmly makes it easy to understand. Thank you so much for the video.

[@eaterofcrayons7991]

Actually goated, love how you break down fancy concepts/rules into manageable steps

[@benishot1000]

omg i have a quiz on this on monday thanks nancy!!

[@pendragon7600]

how did the quiz go?

[@benishot1000]

i guess it went well but my teacher lost my quiz and gave me a 0

[@pendragon7600]

that sucks

[@rebelfriend1818]

Benjy Molina lol wtf

[@Rafayrealityfun]

You are so cute tutor nancy!!❤️i really like your style and be a found of calculus🥰

[@mohammedwasifulhaq6295]

I can’t imagine such beautiful lady can teach math in such beautiful and understandable way Thank you for your great help

[@herobrineslayer1585]

So relaxing and slow paced. Perfect for my learning style

[@christy967]

Thank you so much you are such a talented teacher. It's so relaxed here and not intimidating. I love it

[@luanfigueiredo7283]

THANK YOU SO MUCH. I ONCE THOUGHT I WAS DUMB BUT U CAME TO RESCUE ME FROM THE DARKNESS OF FAILING MATH. I APPRECIATE YOUR EXISTENCE AND WORK!

[@karinadeleon7138]

You are intergreat

Oh for the love of God, *STOP!*

[@realmadridfan9949]

Karina De Leon corny

[@kumarupendra]

kzbin.info/www/bejne/r5C7qIShmM2qqKc

[@abzshaker]

@ god has no place here

[@rakeshkhanna7253]

Thanks for the step by step treatment of integration using U-Substitution which made me understand the subject quite comprehensively.

[@Spluurj1776]

I introduced your channel to my daughter who is taking geometry. Thanks for making a page with really easy to follow teachings

[@rheinnabi5052]

why do people like her not have millions of subs, they're doing more good than rice gum and jake paul

[@danielross5931]

if she dabs she will get famous

[@vincentgrimaldi3652]

Think about the potential audience size. People who are using KZbin specifically for Calc 2 versus people who are using KZbin to consume "entertainment media", the market sizes of those two population groups are wildly different, and thus, the subscribe base is usually commensurate with that. Not saying it's wrong or right, but that's why.

[@abab1247]

@@vincentgrimaldi3652 well it is since humans prefer entertainment over work and studying.

[@sircakey-5902]

it took me so long to figure out how you were writing backwards.. but now i realized you probably just inverted the video lol

[@done2913]

you are one of the best teachers that i can understant in the first example thank you so much

[@Sunnyk6631]

You deserve million of subscriber 🙂😊. it's really helped me to clear my concept when no other videos on youtube available like this . ❤️

[@michaelzacharie7484]

Next Video Suggestion: Calc 3 - Scalar Multiples (Vectors) and Parallel Vectors

[@sansamman4619]

yes please!

[@SWEETG]

Yes please

[@pointlesslylukesplainingpo1200]

I see vectors in my sleep paralysis

[@Grassmpl]

Sounds like you can teach the whole material just via the comment thread

[@saiyanbro]

Genius Nancy said calculus is everyones favorite lol. Greatest teacher and a comedian!

[@NowhereNear42]

One of the best explanations on this topic I've ever seen.

[@XxRespawNxX619]

Thank you as always for the clear and concise explanations !! Wishing you all the success !

[@mark0032]

I think the most impressive part is that she's doing this while writing backwards. I can bearly manage it forwards

[@bakefakecake]

U can write normally and flip the vid in editing

[@unlqsting5475]

if you look reaaaaally closely, you can see that she is teaching math

[@mariamartin5674]

This woman is the reason I passed calc

[@SkippiVarndenkof]

Thank you so much, needed a recap and you've explained it so thoroughly, the technique has clicked again!

[@lydjoy3385]

literally helped me more than my calc professor...

[@luxor5572]

Love how she's laughing at 5:52 because she has covered the pink one from the wrong side.

[@teeyuxuan1896]

you are such a great teacher. You actually made me fall in love with doing calculus. THANK YOUUUUUU!!!!

[@tru2746]

i like how you explain even the smallest changes. the best

[@brandonthomas7698]

the integral of x\/x+2 was on my test and this helped me

[@kawowine]

I love you So much I was so lost (in calc) And now I am found

[@arki3134]

Amazing grace

[@Hei_Darkfire]

I know a lot of people have already said this, but you did a wonderful job explaining this. Even better than my professor. So thanks. I might actually have a chance on my quiz thanks to you.

[@khalolyyan7971]

I Love how you show all the possible formS of problems Thank U

[@kidsconner]

Nancy you are super amazing. When someone is such an expert in Kenya will call him/her Fundi. Fundi Nancy! Thanks a lot!

[@theshadypersonify]

YES.......Nancy I am one step closer to winning a Nobel prize >:) Thank you

[@IMFletch2079]

Xavier Rodriguez v

[@haidermehar2422]

Such an Amazing teacher i had ever view .so polite and so grateful 💞

[@arielllerena8355]

omg a math tutor that actually goes slow, what a miracle

[@ryanluepke9191]

play this at 1.25 speed, you’re welcome

[@stardustlordfinery8140]

1.75

[@erfankamali8293]

thank you sir

[@stifflery]

No

[@Kappi1997]

I don't feel like this. I love that she takes it slowly it makes me less agressive than normal mathteachers

[@benjaminsibson4265]

dhruv SIMP

[@blitzpanzer1016]

Question: how do you write on that. Is that a clear board and if so did you learn to write backwards or something? Im sorry does anyone else have this question too. Like it bothers me just cuz. I know Im weird.

[@SteveGarra]

I think that she just writes normally on the clear board, but then in video editing just mirrors the image so it appears forward for us and not backwards.

[@Defender2002]

Please don't ever take these vid's away they are priceless

[@mayankbanerjee232]

wayy more easy to understand than my teacher. This all finally makes sense now!!! I might actually pass this class now that I've found you, thank you!