Integration by Parts… How? (NancyPi)

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

  2. 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

  3. 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 ;).

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

  5. 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!

  6. 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

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

  8. 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.

  9. 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.

  10. 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.

  11. 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.

  12. 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)

  13. 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

  14. 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!

  15. 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

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

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