Basic Integration... How? (NancyPi)

MIT grad shows how to find antiderivatives, or indefinite integrals, using basic integration rules. To skip ahead: 1) For how to integrate a polynomial with the POWER RULE, skip to 1:35. 2) For how to integrate NEGATIVE POWERS of x, FRACTIONAL POWERS of x, and RADICALS/ROOTS, skip to 6:12. 3) For how to integrate x^(-1), or 1/x, with the LOG RULE, skip to 7:23. 4) For examples where you use more algebra to rewrite before integrating, to SEPARATE the numerator of a FRACTION, or EXPAND a PRODUCT in order to use the Power Rule, skip to 10:00. 5) For basic TRIG and EXPONENTIAL examples that use rules from the Table of Integrals, as well as trig identities, skip to 11:36. Nancy formerly of MathBFF explains the steps.
For my video on how to integrate using U-SUBSTITUTION, jump to: • How to Integrate Using...
For my video on how to do INTEGRATION BY PARTS, jump to: • Integration by Parts.....
Follow Nancy on Instagram: / nancypi
Twitter: / nancypi
1) POWER RULE: If you're integrating a polynomial, or just a power of x, you can use the Power Rule on each x-term in the polynomial. The Power Rule says that for a term that's just a power of x, such as x^3, you can integrate by raising the power by 1 AND dividing by that number (the new number you got by increasing the power by 1). For example, the integral of x^3 would be (x^4)/4. If there was a constant multiplied in front of the x-power, you can keep the constant and integrate the rest of the term (Constant Multiple Rule). For example, the integral of 6x^2 would be 6 times (x^3)/3, which simplifies to 2x^3. You can repeat these steps to integrate every term in the polynomial and string them together for the full integral. You can also keep the addition and subtraction between the terms (because of the Sum and Difference Rules). NOTE: The anti-derivative of a constant (just a number) is always that number times x (the Constant Rule), so the integral of 1 is 1*x, or x. At the end, it's very important to remember to add a constant of integration, to include a "+ c" at the end of your answer, when it is an indefinite integral (integral with no limits). ANOTHER NOTE: The Power Rule only works when the power is not -1.
2) NEGATIVE & FRACTIONAL POWERS, and RADICALS: For negative powers, you can still use the Power Rule, as long as the power is not -1. For fractional powers, you can also use the Power Rule. When you increase a fractional power by 1, you will have to simplify the power (before integrating) by getting a common denominator. For roots, like the square root of x, it's best to rewrite the radical as a fractional power, and then use the Power Rule.
3) IF THE X POWER IS -1 in the integrand, either written as x^(-1) or 1/x, you have to use a special rule, the LOG RULE, that you can find on a Table of Integrals. It says that the anti-derivative of x^(-1), or (1/x) is the natural log of the absolute value of x, plus c.
4) MORE ALGEBRA: Sometimes you may need to do a bit more algebra before integrating, so that the integrand is in a form that fits a basic integration rule, like the Power Rule. If you're integrating a rational expression, you can sometimes separate the numerator, or break the fraction into separate fractions, simplify each term, and then integrate with the Power Rule. If you have a product of x expressions, you can multiply it out, or distribute the factors so that you have just a polynomial (and can use the Power Rule).
5) TRIG & EXPONENTIAL: You can find a lot of trigonometric (and exponential) integral rules in the Table of Integrals. If you don't see it in the table, you may need to use a trig identity first, to rewrite the integrand into a form that you can integrate.
For more of my math videos, check out: nancypi.com

[@justagenosfan]

6:13 my match teacher when I turn my eyes away for 1 second

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[@adrianv.v.4445]

LMAO

[@mujahid4764]

😂😂😂😂

[@nesto9889]

At that point I was like: "Nevermind I'll wait until I actually get this in school"

[@daviddial6147]

Only Nancy could make me watch a video on Integration after she already helped me pass Calculus.

[@Gjonesy9]

I'm so confused!

[@AlinaKay47]

Wow same

[@daviddial6147]

gjonezy Sorry, I am as bad at sentence structure as I am math. I mean that even though I already finished Calculus (Thanks to Nancy's other videos) I am still watching her video even though I do not need to.

[@daviddial6147]

Congratulations Vib, that is no easy task.

[@strangeperson700]

She is a beautiful perfect female specimen of asmr mathematical goodness!

[@shafe9866]

Imagine your math teacher explaining some difficult math stuff with a smile on his/her face

[@dont_touch_my_cupcake2938]

impossible.

[@txan9700]

it would be soooo cringe ...but not her ....her smile looks natural

[@AvenEngineer]

Hi Nancy! I've watched a heap of your videos over the past 3/4 years. I just wanted to write a comment to say thanks. I'll be starting my first job as an engineer in a few weeks. :)

[@mariovanda9504]

.

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Nice! i have just started my 4 year course to train as a B1.1 license engineer, any tips?

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[@TrollerzTV]

With nancy and prof leonard anybody can be ace calculus 1, 2 and 3

[@dsrsk8354]

Thanks for doing my teacher's job and being the sole reason I can graduate- especially now with COVID. I have a 12-hour difference with my online classes and my teacher often forgets to upload his recorded classes/or make them inaccessible, so all your videos have been a lifesaver.

[@bradsmith6019]

No need to apologize for "insulting" with algebra talk. I appreciate that steps and explanations are included. These reminders are helpful sometimes. Another great video . Thanks a lot!

[@SilentBoy]

The math teacher we all wanted

[@iluvfb071891]

Really just wanna say your channel is AMAZING. you explain things so beautifully. It makes the hardest things simple.

[@brookogden1542]

Such a great teacher!! I have an A in applied calculus because of you. Seriously, I really hope she is teaching somewhere so more people can benefit from having an awesome teacher! Thank you Nancy!!

[@racingeye3826]

congratulations mate

[@colorblindexposures]

She teachers here man lol

[@Titoelpadre]

Just want to say thank you for your incredibly helpful videos! I literally find myself walking around my campus at 3:00am cramming calculus the day before my exam. ( And actually pulling it off thanks to you! )

[@angelm2567]

Thank you!!! I was making these much, much harder than they needed to be. (For a year, now.) You and all your videos are greatly appreciated!

[@blessedkutsirayi9262]

Ohh how I wish I had met Nancy a long time ago. She explains very well and she has patience during her lecture. God bless you, Nancy

[@JD-ye2pw]

She's a tough teacher. Will not spoon feed you But push you. These videos motivate me

[@TimHerbert509]

I love it soooo much when I see you covering any topic. You have saved me countless time on my Calc 1 class. Thank you so very much!!!

[@endrankluvsda4loko172]

This is the first video I've seen of yours, and I had to subscribe pretty much as soon as it started. I love the way you explain things in a way that's not too long and drawn out but that is also clear and helps to explain things. When I was in college, I was going for high school education (math), but to become a high school math teacher, you basically have to become an engineer it seems. I never really got the whole Integrals idea. I was so lost in Calc 3 that I switched to middle school education just so I could be done with taking math courses. I've always loved math and wanted to learn it though so I'll definitely be watching your videos and hopefully finally understanding all that stuff I could never get before.

[@MattAkiyama1]

To me it's important to give small reminders of algebra steps because it's a good reminder for a lot of us who forgot how it's done. I appreciate those reminders, very helpful! The people who end up needing your videos are usually ones with weak algebra skills.

[@rudhrapanchal4049]

First, I thought that she was just writing everything backward. And then, I realized that there is something called "mirror the video"...

[@hiredgoon4269]

This is so much simpler than how my calculus professor made it out to be. Thank you. Rewrite. Integrate. Simplify.

[@Invizive]

"You're still alert?" This is how you startle people in educational videos

[@alvinthechipmunk7014]

This was genuinely so good. Thank you so much for making this video. You really, really cleared this daunting chapter up.

[@rodshop5897]

Thanks, this is a great refresher as I get started in Physics. Nice video: good speed, good diction, well explained.

[@shamsunnaharmily2014]

thank you soo much nancy! You explain so well. I had mid term and quiz and all i did is watched your videos to clear the concept and it helped me a lot to pass my exams.

[@LuizMartins1989]

I just need to watch your videos for a few seconds and I get concept!!!! You are 1 in a billion girl! Thank you and keeping making these videos. You are truly proving a public service to students.

[@argynramz7628]

*Thanks Nancy!* *I've bookmarked some of your videos back then and suddenly the videos got removed from your previous channel (mathbff). I just wanted to write to say thank you for re-uploading the videos.* *It really helps a lot* :)

[@angiolg2]

Way better than my teacher or my textbook. Thank you for making it seem simple :)

[@kaylawagner3295]

Thank you, Nancy! Please make more calculus videos. You explain the best!

[@RtsFps1]

It feels so simple when you explain things step by step. Thank you so, so much

[@bacillusrex]

I started an MSc in applied economics recently and these videos are a lifesaver.

[@horlatunde8845]

this has just helped me wipe away my fears on integration and now i can easily do my test without help nor fear....thanks a lot Nancypi

[@gopalsivethna2959]

is this woman writing everything backwards

[@prathameshp3013]

No, she probably writes everything the right way round for her, then reflects the video to get the mirror writing the correct way round for the viewers

[@stardaggerrihannsu2363]

To go to M IT you need to have at least TWO freakish talents and an IQ of 127+ and a perfect GPA. One of her freakish talents is writing backwards!

[@robert-rv8lo]

No, she is right handed.

[@stardaggerrihannsu2363]

@@robert-rv8lo I'm not sure that MIT allows right-handed goons into their Math programs!

[@robert-rv8lo]

@@stardaggerrihannsu2363 Supposedly if you wear lead underpants they'll let you.

[@TheQassim10]

thankx for coming back!!! your videos are really helpful and i have been watching your videos fora while

[@Samopersona22222]

Thank you so much! I never taught I will learn something in math because it’s just too hard for me to understand. After many videos and channels you helped me a lot and English isn’t even my first language. You are explaining way too good and make it easy to understand. Thank you I appreciate you a lot.

[@Dedicated4Tech]

6:06 check phone notification 6:34 board after 30 seconds of not paying attention

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underrated comment

[@BattleKing00]

That's how I feel actually in class.

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😂😂😂

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HAHAHAHAHA

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Thank you from the bottom of my heart.

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[@bharathir399]

wow, you just summed up my last two weeks of confusion and made it seem like nothing! you're amazing!

[@stephaniejoseph8514]

I've never liked a Calculus teacher more than I like Nancy Thanks for the helpful video! 💕

[@HeelBusiness]

Wonderful instructor! I needed a video like this to help out students that I tutor. (I’m rusty - it’s been years since college!)

[@valerianevarez3139]

This really helped me get a better understanding, thanks!

[@itsangelinapeople]

I understood more in 2 minutes than I have in the past two weeks of my class. Thank you, I’ll watch all the ads without skipping.

[@Oliviaismexoxo]

Lol

[@diencai1812]

Thank you Nancy. You made a difference to my learning journey

[@junaidahmad4344]

9:34 "u're still alert?" me: uhhm...uhh...yeah..u pretty asf..uuhhh....what???

[@recondinent2]

The first 11:30 of the video: "oh, this is easy!" The rest of the video: "you know what, I'll just lose these points in the final"

[@jvg7806]

you're about to lose lots of points lad

[@rashedalshebli5806]

and an trying to study them for my final tomorrow and I lost them :)

[@pinpon163]

Omae wa mou shinderu!

[@recondinent2]

update: got a C. ayyyy good enough for me :D

[@hisyamhasbi8257]

@@recondinent2 whoa u study 11 months before the actual test?

[@keronnicholson8313]

I ain't lying..I love you Nancy! Thanks for the lessons.

[@ob4ra]

"You dont have to like math, but you can like my videos." had me.

[@ProtoMaxoid]

A review before I start Calculus 2 thanks Nancy

[@aswin5046]

How do u feel now??

[@Hindiclass.]

I wish u were my school 's maths teacher. Cause what ever my maths teacher teaches it feels like even he himself is confused and standing in front of white board to clear our doubts

[@jbuckmusic88]

Thank you for such a concise and easy to understand video! You might have just saved my grade in physics!

[@bobbybrown2916]

Good morning Nancy,I really love your way of teaching, you actually take your time to explained, you are helping me to understand a lot,may God blessed you.

[@misanthropickryptonian2626]

WOW!!! THAT CLEARED MY 2 YEARS OF DOUBT!!! CHANGED MY YEAR!!! THAANKS!!!

[@vibz123rulez]

Ive already graduated in aerospace eng and im still watching this. I dont know why...guess its good to keep brain awake

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Nice one Patrick, ze is indd wel heel erg knap

[@ramazann9827]

Aerospace? Can I ask questions 😃

[@ramazann9827]

Im a student 12th grade in Turkey and I want to be a aerospace eng.

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@@ramazann9827 Mükemmel, Inşallah sen bır gün olacaksin :)

[@GryffieTube]

Why can no one else explain it like this? Thank you SO much, great video and excellent explanation! Subscribed :)

[@jefferysilerjr2330]

This was so gosh darn helpful I had to pull out another device to like it twice. In 2021 these videos are still top tier.

[@bartholomeuwtitus1461]

I swear Nancy will be the reason I get my engineering degree!!

[@hellothere5350]

Bartholomeuw Titus wait this is what ur learning for degree how old r u? I started this properly when I was 15 granted a bit earlier than most would have at school

[@MrGriff305]

She's an android math bot. Every once in a while, she engages fast forward.

[@tigertrophy1805]

Nancy, You are a left handed genius. You make this math look easy. Two formulas only per page of notebook.

[@shardonayM]

thanks for the help, someone put your videos in the group chat and you are a blessing.

[@mtm7862]

Thank you Nancy

[@lost3263]

14:38 thanks for that little heart over the 'I' of trig to make us feel positive beside our hate against trig 😂 anyway I love ya teaching 😁✌🏻

[@zulyc8641]

your videos helped me so much with calculus (got a B, which is better than I thought I could ever do) and now im using them to review calc for calc based physics :) thanks so much!

[@leno__2279]

As a fellow lefty the amazing thing about your video Nancy is that you are left handed writing backwards on a dry/erase board. That’s real genius right there.

[@MrSalman108]

A lot of students give up because no one tells them or convinces then why they need calculus +nancypi Maybe can do a video about that ?

[@heaintloveu]

Because in order to earn a degree you need to be able to do upper-level math

[@harjitsingh7308]

Calculus is found in alot of palaces. It is found in physics(when studyig about gravity, thermodynamics, quantum physics), it appears in business(when studying about prices and other models), it appears in geometry, computer science, biology everywhere

[@harjitsingh7308]

Depending on what role you do will depend on whether you actually use calculus. For example, if you are a web designer you won't even touch calculus but if you are working in AI/machine learning you will utilise alot of calculus/linear algebra/statistics

[@markspazzmaster388]

This doesn't help. I don't know if I'll work in a palace after college

[@wranther]

OMG Nancy, how did you do that? Just a wink and the board cleared for you, just like Samantha Stevens on Bewitched!! LOL

[@Inquire98]

I LOVE Calculus 😉 Your presentation/video makes Calculus more entertaining and understanding 😎

[@karthikm2727]

Nancy I really thank you for your effort

[@sub.umeshprasad8639]

*You really deserve more subscribers*

[@polincercleron6179]

Nancy: you're still alert? Me on my phone: oops Nancy: That's Amazing!

[@bunkydunk7500]

I am a new fan. Your videos are very helpful for me in Calc 4 as i need to review a lot. Also it helps me be a better tutor 😊 You're amazing!!!!!

[@jamieramshaw8425]

Literally doing a chemistry degree and these videos are a lifesaver

[@Dria.Unedited]

can you do a video finding the area under the curve next (like using right and left endpoints of rectangles)

[@bollywoodsongsatoz4262]

Madam please upload video for double and triple integration and curve area and vector integration

[@Sneaky1ne]

sinx/(1-cosx^2) can be solved with power rule, I had a nagging doubt that I could do it like that so I had to get out of bed at 2 am to try it, because math is awesome

[@elmerbernardo5213]

i can listen to your voice for hours. thanks for doing this :) the world is a better place :)

[@charlielamme4256]

Thank you, my highschool teacher has no clue how to teach a class and I have a test tomorrow. I asked her for help and she told me to get a tutor. Why would a teacher tell a student who is struggling with the material that. Only she knows, anyway you are a legend. Thanks

[@gowthamkrishna1568]

"You're still alert!?" Bold of you to assume.....

[@verenameli9095]

Seriously? I find this video in my recommendations not even 24 hours AFTER my integrals exam?

[@yellit69]

Thank you so much, I'm a returning student and currently using your material to study electromagnetics in uni... thank you so much

[@benny6458]

This is so helpful. I'm starting this topic tomorrow after the Spring break.

[@Mr6Sinner]

“I know calculus is exactly what you wanted to be doing right now” Oddly enough, yes lol

[@matthewfuerst6456]

Uriah Siner lol same I’m here cuz I’m actually curious about integration. Also cuz I’m procrastinating about writing an English essay

[@yobnok]

6:14 Alright i'm dropping out of college

[@VeritasEtAequitas]

Many people do trig and begin calculus in high school. :(

[@sumnickd]

dude same

[@Vikermajit]

Thanks again teacher Nancy, your videos on calculus are beginner friendly, systematic and explained at a moderate pace, each to follow....tqvm...

[@yasserkhreis9605]

Thank You Nancy for your explanation

[@francesmcbride5936]

Thanks for the vid, Nancy! I was so sad when MathBFF stopped making videos, but I'm so happy to see that you're making calc videos again. I loved some of those old videos about advanced integration techniques that have sadly been taken down. Do you think you'll be redoing those any time in the near future?

[@usamarafieck1715]

Is this calculus 1?

[@amirtorhan2762]

i had to think for a solid 10 seconds if she was actually writing backwards this good

[@mryup6100]

Lol

[@robertdesanctis1238]

Kids today are so lucky to have KZbin to help with all subjects. Nancy is great. I loved calc 2. Learned integration from dr kosmalla.

[@presterjohn7789]

Wonderfully, cleanly spoken. I felt like you could have uttered an uh or two there, but you didn't. Impeccable presentation.

[@patricknkwo652]

All through this video, i asked myself how she writes inversed.

[@hardikvegad3508]

Every time she makes an eye-contact...My heart literally skips a beat.

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that's just your arrhytmia

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@@namanbhatt6046 i wish it was the case... But unfortunately it's not

[@namanbhatt6046]

@@hardikvegad3508 it was a bbt joke dude

[@lostastros3039]

thank so much, I just needed a refresher from when I took calc 2 so long ago and now I regrettably needed integration again in physical chemisty

[@Elz_R]

Your videos are helping me a lot! Thank you so much!!!

[@brcarter1111]

"Let me show you how do do it".... *****stops talking, extremely serious face, fast tracking while writing formulae, smile*****

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Accurate af

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huff ... still prefer her more than you hehehe

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@@abcdef-py6rh roasted

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If she was my College professor, I wouldn't miss a single class

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I Love math and I Love your videos. This is what I Love doing on a Friday night!

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Maaaannn that's clever. The bit about the conversion from 6 to 6x to the power of zero would have never crossed my mind. That is good !!!

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Nice vid ma'am !

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So ja nind NAHI ari kya

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@@abcdxx1059 lol

[@AS-ey3vg]

u make me like maths wtf, u saved my exam btw

[@TheEndernal]

How cruel of her to make you like maths lol

[@kashifaakhtar6510]

Mam, Only because of you I really understood Calculas. I am very thankful of you... Plz keep making videos of other chapters too.

[@natealemagno1369]

I LOVE all of your videos and they help so much!! but now that im in Calc 2 its a lot harder to find the information and help that i need! pleeease make more videos soon!

[@jesussaquin6266]

6:15 yep ill just switch to arts degree

[@thepunisher3677]

Dude this is just super basic level stuff lol

[@jesussaquin6266]

@@thepunisher3677 ik its meant to be a joke.

[@user-fr9ec4xq1e]

The Punisher leave me alone not everyone is a math genius

[@sarakingg15]

Can you do a video on Riemann sums and definite integrals?

[@AlanCanon2222]

I'm 51, a former physics undergrad and in corona virus lockdown, Calculus is exactly what I want to do right now. Great lecture.

[@mrch33sehed93]

I remember when this was sooo confusing. Your videos are great at refreshing my memory and reigniting my admiration for math. Thanks Nancy 🙂👍