Hi guys, I’m Nancy. I’m going to show

you how to do integration by parts. So you’ve probably been integrating for a

while now, but it’s like a zombie. You thought you had integration down, but now

it’s popped back up again and now it’s integration by parts. So let’s do it. So

take a look at this. Say you have to integrate something that looks like this.

How do you do it? Well, first of all, ask yourself if maybe this is something that

you do know how to integrate already right away just by looking at it. I don’t. I

mean, if this were just integral of x, dx, we could do that easily. If it were just

integral of e to the x, dx, you could do that with the basic integration rules,

which hopefully you’ve learned already. But it’s not that. We have the two of them

multiplied together. We have this product here, which makes it trickier. So what do

we do? Well, first of all, quickly you should check to see if maybe you can just

simplify this with algebra, combine it, and put it together. You can’t do that

here. I’m just saying it, because sometimes you can and that would make it a

whole lot better, but that would be too easy here. So what do we do? The next

thing you try is a substitution. So you use substitution. You should try it, see

if it works and helps. I’m telling you that no matter what you chose for u, for

substitution is not going to make it integrable, like if you pick u to be x,

your du won’t be what you need to cover the rest of the integrand. If this were x

squared, then a u there would help because this would be what you want for your du,

the right order, the power of x, but it’s not. But you should try substitution

first. So then what do we do if all of our usual tricks won’t work? Well, we need

something new. This is the integration by parts formula. What is that? All it does

it takes your integral, and it rewrites it using a new, different integral. Why would

we do that? Because hopefully the new integral is easier to do, something we can

integrate. So a lot of the work of integrating is whipping something into a

shape we can actually do it. So this sounds great. Wait, what’s the catch? You

have to choose what’s u and what’s dv yourself and carefully in order for it to

work. So choosing u and dv can be the most confusing part, the hardest part I think.

Don’t worry, I’m going to show you how to do it, and I’m going to show you a trick

in a minute, the LIATE, L-I-A-T-E acronym trick. But first, here’s the general idea,

a good rule of thumb. And you’re going to see a lot of different rules and mixed

messages out there, but generally for this whichever of your two factors gets simpler

when you differentiate it, make that your u. And then the other one will be your dv,

unless that one would get more complicated when we integrate it, which doesn’t happen

a lot. But basically, pick for u whichever of your two factors gets simpler, breaks

down, gets smaller when you take the derivative of it, so maybe the degree gets

smaller, the order of it gets fewer terms. If it reduces in some way, make that your

u and the other one your dv. And if this all sounds really confusing, that’s

because it is. So let me show you what I mean in this one. Which of our two

factors, x or e to the x, gets simpler when you take the derivative of it? Well

the derivative of x is just 1, that’s simpler. The x dropped out, it’s a lower

order or degree, or power. The derivative of e to the x is just e to the x again,

which is not simpler. So probably our u is going to be x, and then the other factor,

e to the x, should be our dv, unless e to the x will get more complicated when we

integrate it, which is doesn’t because the integral of that is just e to the x, which

is not more complicated. So we’re not in danger of that. Our dv is just the other

factor e to the x. And this is important, whichever one you pick for dv also gets

the dx, because you need that differential there. So this is great. We have our u and

our dv. Now we can plug it all into the integration by parts formula, but first we

do need to find two more things quickly. We have to find du and we have to know v.

Okay, so we’re going to differentiate u and we’re going to integrate dv. So we’re

going to get du by taking the derivative of u, and we’re going to get v by

integrating dv. So if you take the derivative here, we get du by taking the

derivative of x. Derivative of x is just 1. And anytime you’re getting a du or a

dv, don’t forget the differential at the end, the dx. You need that, so don’t

forget that. And then if we integrate to get v, integrating e to the x is just e to

the x again. So v is e to the x. And just in case you’re wondering, in case you’re a

clever one, you don’t need a + c in this case. It won’t matter in the end result,

so you don’t need to worry about writing a + c at this point. So now we have

everything we need for the integration by parts formula, so we can plug in to the

formula. Okay, so now we’re going to use this formula on our integral x, e to the

x, dx. X is u, e to the x, dx is dv. And the first part of the formula, what this

becomes is first u * v, which for us is x * e to the x – the integral of v * du. And

we have those. V is e to the x, du is 1dx, so this is going to be minus the integral

of vdu, which is e to the x, 1dx. And we don’t even need to write the 1. This is

the same as dx. The one is implied, so it’s going to be minus the integral of e

to the x, dx, okay? So we’ve used the formula. I know this looks like it got

more complicated, but it’s going to work out, and we’re almost done because this

integral is something that we know how to do. This is an integration rule, the

integral of e to the x is e to the x. So this is what we have when we integrate

that. And at the very end, don’t forget to add a + c. This is an indefinite integral

that has no limits here, so you do need to add in the constant of integration at the

end. Don’t forget the + c. So this is our answer for this integral. So integration

by parts made this solvable, made it integrable like magic. And if you want,

you can check this answer, you can take what you got, differentiate it, and you

should get back your original integrand, x, e to the x, and you will. So in

summary, all you need to do is pick u and dv, find du and v from them, and then use

the formula. I mean look, guys, if you pick the wrong u and dv the first time,

don’t panic. It’s okay if doesn’t work out the first time. If it turns out that your

choice doesn’t actually break it down and help you integrate, you can try something

else. No harm, no foul. I have done it and part of getting good at something is

bumping up against what doesn’t work. So the more practice you get with it, the

more skilled you’ll get at picking u and dv. Ah, but you say, “What if you hate

skills and hate developing skills, and you would rather have a blind Rote trick

handed to you on a silver platter that always tells you what to pick for u for

sure?” Well, then I have just the thing for you. So here’s our trick. It’s an

acronym to help you pick u. So u will be the first thing you find in this list of

letters. And dv will be the next thing you find. L stands for log, so that could be

natural log, lnx, or normal log, log. I for inverse trig functions like arc sine

x, cosine inverse of x. A stands for algebraic or polynomials, powers of x, x

squared x, x cubed. T for trig functions, like straight up sine, cosine. E stands

for exponentials, like e to the x. So the trick is to follow these letters in

sequence. And the first one you find that you have is your u. So what do we have? We

have x and e to the x, x is algebraic. E to the x is exponential, so we have A and

E. If you follow those letters in sequence, the first one you encounter that

we have is A, algebraic, so that x has to be our u. U is x, and the next thing we

find that we have is E, exponentials, so that’s our dv. Dv is e to the x, dx. So

that’s the trick. It’s a fun party trick. Now let’s really put it to the test. All

right, look at this integral situation. Because I think the hardest part is

picking u and dv, here are just a bunch of different types, a mixed bag of types you

might see. It’s a real mess. So let’s talk about it. If you look at this one,

integral of x, sin x, dx, we can use our trick. We have x, which is algebraic, sine

x, which is trig. Since A comes before T, this one is our u, and this our dv. Dv

also includes dx, remember? So whatever you pick for dv also gets the dx, just

like up here. By the way, if you didn’t use the trick, it still works to think

about it like how we were saying before. Whatever you pick for u, you want to get

simpler when you take the derivative, which it does. And whatever you pick for

dv, you don’t want to get any more complicated when you integrate it, and

this doesn’t really. So that’s why. But you can stick to the trick. These other

two are kind of the same form, algebraic and trig, algebraic and trig, and you can

see that the algebraics won out. They got placed as u because they appear first in

our trick. And then these are dv. Only thing I will say is that in this one,

you’ll need to do integration by parts twice. Yeah, I know. That’s a whole other

video. But it’s exactly what it sounds like when you do integration by parts and

you get a new integral. You do integration by parts on that integral in its place.

And if you do that, you will get the answer. But anyway, let’s look at this

kind. This ones here, because I don’t want you to think that any time you see an x

term, that that’s going to be u for sure, because sometimes it’s not, like here.

Here we have algebraic and a log. And following our trick, since L comes first,

our log will be our u, and the other part, x cubed, dx, will be the dv. This also

goes back to what I was saying, you don’t want something to be dv if it’s going to

get more complicated when you integrate it and lnx definitely does get more

complicated and more terms when you integrate it. But you can stick to the

trick as well. What about this form? We have algebraic and exponential in all of

these, algebraic exponential. Well, since we have A and E, A wins out and all of

those algebraics are the u’s, and the rest or the dv’s. This one you will need to do

integration by parts twice. I can see the future. It’s a very bright future in which

you need to do integration by parts twice for x squared – 1 * e to the x, dx, that

integral. What about this type, exponential and trig? I haven’t marked

anything for this. To be totally honest, exponential and trig, E and T, these are

actually interchangeable. I lied. This could be ET, it’s just that it would be

hard to pronounce. Instead of LIATE, it would be LIAET. So it’s written this way

so you can pronounce it, but really if you have only a trig and an exponential, you

could pick either one to be your u and you will get the answer. It will work out.

Also, you’ll need to do integration by parts twice for that one. And one final

kind. If you just have one term in the integrand, lnx, or sin x, turns out you

can use integration by parts on something like this because the dx can be thought of

as 1dx, and can be thought of as an algebraic term. And if you do that, like

here you’ll have algebraic and log, so log will be your u and the 1dx will be your

dv. Our x sine is an inverse trig, the I in our trick. It comes before algebraic,

so that’s the u, and the 1dx is your dv. So that’s integration by parts. Just a

couple things, if you ever see a definite integral with upper and lower limits where

you need to use integration by parts, you can do it. It’s just a little more work of

evaluating the limits for each term. So in the formula, the uv term, you’ll evaluate

limits for the integral vdu will have limits on it. If you ever want to derive

the integration by parts formula, you can do that. You take the product rule and you

integrate it and it’s not that bad. That’s a whole other video though. Also, there is

a version of the formula, integration by parts, that’s messier looking and it has

f(x), g(x), f prime, g prime. It’s not as easy to use. The kind we have that we’ve

been using is neater and more compact, and it just came from a few substitutions from

the other one, but just know that it’s the same thing. And that’s about it. If you

have an indefinite integral and you’re doing this, don’t forget the + c. So I hope that helped you

understand integration by parts. I know calculus is exactly what

you wanted to be doing right now. It’s okay. You don’t have to like math, but you

can like my videos. So if you did, please click like or subscribe.

Thank you, Nancy!!!!! I have exams tomorrow and thanks for this ♥️♥️

You explain this topic so well. Not only do you enliven the maths but your goodness shines through and you have beautiful arms.

after years i did come back here to recall my DE lol it has been touched with our new lessons so i need to learn it again HAHAH by the way! nancy you look so gorgeous. Beauty and brain be like nancy <3

Butsimplieriscompletelysubjective

am crushing on her

Ok wow

You have shown me how difficult math really is, and have helped me in my decision to drop out of high school. I want to thank you for your help ;).

I'm.trying to remember the proof of this process .. been a few decades of years ..I'm trying to intuitively think about it

OG mathbff!! Only the real ones know

I THINK ITS I LATE AND NOT LIATE

I’m just confused if she’s writing backwards or if she reverses the footage. I can’t focus.

Revisiting from differential equations for a refresher. Feels like a million years ago that you got me through the first half of my calc series!

👍That’s right

I like tricks they make my studies more easy..Thankx nancy love u

nice mam

why would you use by parts on the last one if you don't have to

can we have date cause I like you so much and I do not understand you cause your eyes are significantly beautiful

nancy thanks , your teaching is very accurate , devash , student in analytical chemistry

Thank you Miss, you really helped me a lot in my Calculus.

Are you single?

So hard not to pay attention😍

I was trying to concentrate so hard. But honestly, her perfect smile just got all my attention:C I'm gonna fail

Nancy could you solve difficult ask please

her mouth so big for a kiss!

your tutoring is very awesome.ilike it

When I searched integration by parts on youtube only Organic Chemistry Tutor and NancyPi came with highest reviews both helped me a lot thanks so much Nancy

Inverse trignometric comes before logarithmic….its ilate not liate😊😊😊

Claculus made easy ………….. The best tutoring vedios

Wish u were my teacher

6:47 +c if you are clever one. Haha

Nancy, if wish you were my teacher for every class while I was at school.

LIATE

LIATE

LIATE

LIATE

LIATE

LIATE

LIATE

LIATE

Love you Nancy😋😋you are perfect.. Teacher.♥️♥️. If you would have been my maths teacher, I would have topped my every semester.

thx Nancy

Jesus loves you

Believe in him and repent

Thanks Nancypi! That video helped me remember the process for integration by parts!

Hi Nancy. You have made me fall in love with math once again. I was almost giving up on college math but now, I"m seeing myself doing much better in Engineering Math.

Thank you very much it is very useful

Thank you so much for this!!

Ok here is what I am struggling with. Are you writing backwards? or are there two cameras? This is bothering me.

I'm gunna marry you one day.

you made this concept a lot easier but now i am hard

Am i the onky one who thinks she looks like Megan Fox?

Learnt more in the first 3 minutes than a 2 hrs lecture on this

very helpful…… thanks alot

A beautiful woman doing math?! Is this real?! Is she real?! This can't be legitimate…. Is it?!

Mr Sathia sent me here 😉 Great video by beautiful lady.

I'm from Brazil, yeah im crazy enough to learn integral in a different language im used to.

Thanks that's helped me a lot.

Beauty😍💓

This video helped me a lot! <3 Thanks!

i think i love you!

Please Stop Smiling bcz i cant concetrate

Want to text me before my final lol

I got 2/3*e^x^3 +c

Nancy burn my 💓 heart.

I love u sexy math lady

If Nancy replies to this comment in a nice way I'll eat my boxers

Im not even taking any math classes

i spent the first 20 mins of the video staring at you

i learned alot

You are the best teacher

this was in my calculus 1 final….

8:58 math is not magic. She is magic and looks Like magic

Love u nancypi

How could she be so beautiful and so smart? Is that even legal?

She’s bomb and cute. Whoever gets there is hella lucky

I think it is ILATE, not LIATE. Just to be sure I googled too and found that it is ILATE. All webpages showed that it is ILATE and only a few shows its LIATE. Just a lil bit confused. Correct me, please.

Thank you. i'm doing some review for my calc 2 class next semester. One thing I would recommend is a time stamp or something so people who are refreshing for review don't have to sit through the explanation. Maybe it's just me but when I can't remember and i'm watching a video that 2-3 minutes of "this is why we do it" or "why do we do it" is like nails on a chalkboard. Some need it, maybe even most need it. But i'm sure i'm not the only person just doing a refresher.

なぜオススメに…

Hi

Can u please solve by the above method

Integrate x(4x-1)^(1/2)

Damn

Mire ha mi yo tengo halgo ke exponer sobre Hoyente ha distansia local y despues de haser el matematicas y Ganar (cual es la de Doctor sera igual a la de no DECADENSIA)

I've been trying to find an informative video for so long and this made so much sense, thank you!

You so beautiful that I couldn’t understand anything

Will you do a video about partial fractions?

I learned everything i know about calculus from YOU, i'm so so so thankful, and you're so good to listen to:D keep on doing this, because you're a treasure and the saviour of students<3 so THANK YOU

Cutee and intelligent 🙂

Your teaching is impress

So beautiful yourself and beautifully explained. Lovely

So beautiful yourself and beautifully explained. Lovely

So beautiful yourself and beautifully explained. Lovely

Plz lec on the partial fraction

Lec on the partial fraction ????

Please make more videos.

Is she writing backwards?

Hi Nancy, can you factor out xe^x-e^x+C to be e^x(x-1)+C?

She is too hot for a teacher

I would have attended all my lectures if she was my teacher

Nancy Pi, these videos are just amazing. I really hope you find a career in being a professor one day…education does NOT get any better than this!

You are amazing.

Hi guys I'M nancy 😀

I like nancy is beutifoul

Thank you so much Nancy!!

Your very has helped me to revise before my exam and actually have the guts and relaxation to study. What I mean is that you've explained the content so well and smoothly that I forgot my math anxiety

Worst lecture thooo

Hi Nasi, what is the name of the program that you use to cut videos?

Lessons and Kim are very useful and I would like to give lessons in Arabic like this

I am a mathematics teacher.

My college professor is very cute, so I was unable to concentrate. You are also very cute, so I am unable to concentrate again.

Keep teaching Nancy, please!! Love you!! 😉

up load the lec on the partial fraction

damn nancy's really hot

This is cakework copared to what my professor has been giving us and its only the first week of the semester😣

i like saying it like LI-EEEEEE-AT

11:49 My brain during an exam