## Integration by Parts… How? (NancyPi)

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.

## 100 thoughts on “Integration by Parts… How? (NancyPi)”

1. Shermane Matas says:

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

2. 634k1 says:

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

3. Rachel Ji says:

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

4. Avdhut Mahadik says:

Butsimplieriscompletelysubjective

5. Wisdom Munangisa says:

am crushing on her

6. Melissa Moussa says:

Ok wow

7. Lasse Nielsen says:

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 ;).

8. The Kaveman says:

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

9. Rele Lamola says:

OG mathbff!! Only the real ones know

10. Saswat Padhi says:

I THINK ITS I LATE AND NOT LIATE

11. Daniel Berry says:

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

12. J R says:

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!

13. 大王叫我不要来巡山 says:

👍That’s right

14. football litious says:

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

15. Gullu Shukla says:

nice mam

16. Gabriel Ramirez says:

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

17. ahmad abu qabah says:

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

18. Devash Rampadarath says:

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

19. AungKK says:

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

20. ComedyChristiangamingchannel says:

Are you single?

21. pahul79 says:

So hard not to pay attention😍

22. El Mítico says:

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

23. Mücahit Yeksoy says:

Nancy could you solve difficult ask please

24. gol goli says:

her mouth so big for a kiss!

25. nenshra mella says:

your tutoring is very awesome.ilike it

26. Muhammad Rafay says:

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

27. adrian smith says:

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

28. ASR MURTHY says:

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

29. ItssAlan says:

Wish u were my teacher

30. Parvesh Taneja says:

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

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

31. Yash Kapoor says:

LIATE
LIATE
LIATE
LIATE
LIATE
LIATE
LIATE
LIATE

32. Abhinav Singh Jalal says:

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

33. Balaporte Jean says:

thx Nancy
Jesus loves you
Believe in him and repent

34. Beau D says:

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

35. Marion Ronoh says:

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.

36. Rifat Amin says:

Thank you very much it is very useful

37. K Louise says:

Thank you so much for this!!

38. Zach Jaeger says:

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

39. Kaleb L. Estridge says:

I'm gunna marry you one day.

40. Kevin Kenta Muramatsu-Ng says:

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

41. Lee Roy says:

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

42. Erik V says:

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

43. Clarence Mk says:

very helpful…… thanks alot

44. Vincent F says:

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

45. XuenCheng Sim says:

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

46. Carlos Gonzales says:

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.

47. indian first says:

Beauty😍💓

48. Ferdine Reese Mand says:

This video helped me a lot! <3 Thanks!

49. Parker Gronlund says:

i think i love you!

50. The Silent Genuis says:

Please Stop Smiling bcz i cant concetrate

51. To Jad says:

Want to text me before my final lol

52. Tarek Al-droubi says:

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

53. Rajat Singh says:

Nancy burn my 💓 heart.

54. The Happipotamus says:

I love u sexy math lady

55. Captain Gilligan says:

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

56. Ivan says:

Im not even taking any math classes

57. The mixer says:

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

i learned alot

58. mukarram butt says:

You are the best teacher

59. Garrett 88 says:

this was in my calculus 1 final….

60. Henry Trujillo says:

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

61. Dildar Ahmad says:

Love u nancypi

62. Ifamunas says:

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

63. Daniel Moreno says:

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

64. Parth Gaming says:

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.

65. jdzspace33 says:

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.

66. Boy Mike says:

なぜオススメに…

67. Zehra says:

Hi
Can u please solve by the above method

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

68. Edward Yeung says:

Damn

69. Angel Morales says:

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)

70. Amy says:

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

71. Pew Pew says:

You so beautiful that I couldn’t understand anything

72. Ferenc Fritz says:

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

73. Murat Coban says:

Cutee and intelligent 🙂

74. Nivedit rai says:

Your teaching is impress

75. Hossein Azizi says:

So beautiful yourself and beautifully explained. Lovely

76. Hossein Azizi says:

So beautiful yourself and beautifully explained. Lovely

77. Hossein Azizi says:

So beautiful yourself and beautifully explained. Lovely

78. usman saeed says:

Plz lec on the partial fraction

79. usman saeed says:

Lec on the partial fraction ????

80. l kern says:

Please make more videos.

81. Jonney Silver says:

Is she writing backwards?

82. Emma Holliday says:

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

83. Teodor Yosifov says:

She is too hot for a teacher

84. Bilal Azam says:

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

85. Ryan C says:

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!

86. Ghalib H. Kazmi says:

You are amazing.

87. Md Ghufran Alam says:

Hi guys I'M nancy 😀

88. Les chutes de la mord Les chutes de la mord says:

I like nancy is beutifoul

89. Pedro Mateusz says:

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

90. Muralikrishna V says:

Worst lecture thooo

91. akeel Mohammad says:

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

92. akeel Mohammad says:

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

93. akeel Mohammad says:

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.

95. Pedro Sabino Gomes says:

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

96. usman saeed says:

up load the lec on the partial fraction

97. Calvin Rice-Besse says:

damn nancy's really hot

98. EccentriChild says:

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

99. Richard Richardson says:

i like saying it like LI-EEEEEE-AT

100. Joshua Bourton says:

11:49 My brain during an exam