so finally I'll write down the answer for you guys ladies and gentlemen this right here
done okay this is going to be my last video on convincing people please use the DI method for
integration by parts because they are really the same and I know a lot of people they don't like
to use the DI method because they think that the DI method is just a bad shortcut to integration by
parts and the students wouldn't learn anything by doing the DI method but the truth is the
DI method is just a much better way to set up integration by parts problem so I'm going to
show you guys how exactly I teach Integration by parts from scratch and then hopefully by
the end of the video you will start to believe in the DI method as well so here we go firstly
for integration by parts it's just a way to undo the product rule for the derivative right so let's
take a look at if we have to differentiate the product of two functions I'll just call them
to be u and v and the product rule says I will keep the first function and then multiply by the
derivative of the second so that would be DV and then we add the second function and multiply by
the derivative of the first so this is the classic derivative product rule and now I want to get
rid of this derivative by integrating so integral here integral here integral here this is cute they
canceled but you see that we have two intercourse on the right hand side that's not what we like
I want to move one of the Intercourse to the other side so I can isolate just one integral one side
and traditionally we move this one to the other side so I will just subtract the integral d d u
subtract the integral of b d u so they cancel and I would like to write this down first we have the
integral U DV equals this which has no integral just UV and then minus the integral of v d u
and this is the so-called integration by parts formula the traditional UDD setup and let me just
give you guys an example real quick let's take a look at the integral of x squared times L and X
take a request integration by parts and the truth is anytime you have X to some power P times L next
using the question by Parts but suppose I don't know so if we take a look at this integral if
you look back here the idea is that we want to break down this integral into two parts
one part to be differentiated consider you want to get to Du so how to differentiate
you and then here we have DV and this right here has v so we want to integrate the DV
we have to break this time to two parts one part to be differentiated and the other part will be
integrated so let's do the labeling I'm going to put down the U right here and then DV right here
so what should we do though let's think about it always try to choose this one first because
it's harder to integrate do you really want to put lnx right here to be integrated no because
to do this we actually have to use integration by parts so no [Music] and what style said it
seems that we have no choice but to differentiate L and X so let's put that for the U and I will put
x square along with the DX for the DV so 30 are like this the traditional U TV setup and I will
have to get to Du so we take the derivative right here and either of the L next is 1 over
X DX and we will have to integrate both sides so we can get to V and the integral of this is just
one third x to the third power don't worry about the plus C just put that down at the final answer
so this is the preparation and now check this out the formula says the integral of UTV
we have the U right here and then the DV right there that would be equal to UV which is just the
product of this and that which is this diagonal all right so let's go ahead and put that down and
now write this down first we have one third x to the third power times L and X very nice huh and
then check this out we are going to subtract the integral and the inside is U times d V times d u
which is what just the product of this row okay so you see we have the v d u I'm just going to
write this down first we have the one third x to the third power times 1 over X along with DX and
the beauty of this right here is that the product of this row will give us this integral that we can
actually do because we can simplify this we get x square integrate that but here once their X plus
their power times one third so I'm just going to write down the final answer for you guys one third
x to the third power L and x minus one over nine X to a third power and then we are all done plus d
I call this the second stop in my previous videos where if you can integrate a product
of row you start and just work it out usually you might just need some basic use up or maybe
reverse power rule like this so that's it and now let's talk about what's the di method though
here's the deal if you take a look right here again one part is to be differentiated the other
parts to be integrated instead of putting down U and DV could have just put up D and I I'm just
going to put one part to be differentiated and the other part to be integrated sure right okay
good understand I'm going to put on plus minus to set up why because each row is an integral
I'm going to differentiate our next just about it earlier and I'm going to integrate x square
so you see this time star is the original integral and it was a pass the integral
what do we put on the minus because we have the minus integral here
yeah and now check this out I'm going to differentiate Ln X
this is so wonderful because when you have D that reminds you that we have to differentiate our next
if we have U here that doesn't remind me anything use substitution no not here
yeah differentiate Ln X we get 1 over X that doesn't really right integrate x squared we
get one third x to the third power that's exactly the same right here and then what you do is look
we take this times that that's the first part of the answer and then we multiply the second row
and we multiply the second row we put that inside of the integral
and that's a minus integral from the formula that's it and I know you guys
have seen videos like this on Instagram yeah I make that of course where I just
do the DM method in like 15 seconds and it looks so wonderful but it is just integration by parts
before I show you guys the next example let me count this against this real quick
all right check this out you see here I put on you here I put on TV
and then the next row we have du we differentiated that and then here we have V because we integrated
that and now if I put down the D right here and then I right here and then a plus minus here
you see UV UV times [Music] this thing two things together subtract the integral of this times that
UV minus the integral of ddu is exactly the integral of UTV is that so this thing right
here it should be enough to convince people that they are equivalent but now let's really take a
look at how we can really just do integration by parts if we keep on going for a few more time so
this intercogen means 6 million views on Instagram right now I will actually convinced you guys that
the diameter is truly legit check this out I will do it the traditional way first you and DV
earlier we pick x square to be integrated but if we do that right here we will get 1
3 X Plus third power and that means we will have to differentiate cosine X and then the
DU here is going to be negative sine X and if we do this times that that's actually the worst
integral than the original one the power is bigger and we still have trig function
no that's not what we want with one being said we'll just kind of switch the choice for umdv
and hope for the best it's the best let me show you we are actually going to integrate cosine X
and I'll put down the DX right here so this is the DV part and we'll be differentiating x
squared and now check this out differentiating this we get 2x DX and then integrating this I
will just write this down integrate in your head and we get positive science
don't put on the plus C and now this right here tells us that we take this time step we will get
x square times sine X and then we multiply this two things here and you have to remember that
it's a minus integral and we put these things to get inside and let me write this 2x times sine X
and we have that DX here but guess what this integral request integration by parts again so
let's go ahead and do it so I will pick U and then the V U is 2X and DV is sine X DX why because when
we differentiate this we'll just get 2 DX and when we integrate that we get Negative cosine X
and you see this times that it's a much easier integral than this one and then this one was
easier than that and so on so but anyway we'll just proceed but make sure that you know this
is minus and then we put parenthesis we take this times that we have a negative
a lot of negatives already that's not good it's dangerous right anyway this is 2x times cosine X
and then again minus the integral of this times that so we will have a 2 here and then
let me just write down negative cosine X like so and then we have the DX here
wow so I will close the parenthesis right here good this right here is just a positive
two integral cosine X but let me tell you I'm actually going to use the integration by parts
again here one more time so take a look right here I am going to just put U to b 2 and then
DV to B negative cosine X DX and d u is equal to zero that derivative of 2 is 0. don't put
don't worry about the X and then right here V equals the integral of that is negative sine X
this times that gives us the last part of the answer and again this has a minus and then
parenthesis for this integral and we have two times negative size so negative 2 and then sine
X and finally when we multiply these two things from this row zero times
anything to zero so whatever integral 0 is actually a constant so that's pretty much it
but the problem is that we have to just make sure that we keep track of
the past design negative signs and all that right which is quite complicated
I'm going to start right here because let's take a look at the DI method so check this out D and I
I'm going to alternate the sides on the side how many do I need maybe I don't know maybe
I'll just put down a few just to get ready why do the signs alternate because look
here originally we have the minus because that's from the formula right integral of u d d equals
UV minus integral of v d u that's why we have the minus right here and also right here and why do
we have a plus well you see that this integral requests integration by parts and we will have
another minus integral right this minus times this minus give us the positive right so it's
minus times minus the sign alternates and the next one is this minus here but instead
of another one and good luck with keeping track of those times if you have to do the UTV setups right
crazy stuff much better here and now I'm going to differentiate x square and then integrate cosine X
and we are going to do the things that got me six million views on Instagram I'm just going
to differentiate this one time right here and also write integrate this one time and notice
this is just that same thing integration by parts much easier setup and then if you look at this row
negative 2x times x is this right here and this integral requires integration by parts again
so we do it again and that's exactly this setup and you see this rows redundant because we wrote
it down twice if it's a diameter we just need to write it down once and I am going to
differentiate this again with K2 integrating sine we get Negative cosine X so these two
rows is exactly these two rows and you have the negative right there if you would like yeah okay
the reason I use integration by parts right here again is because this is what I call the
first stop where we see the zero in the D column because when we differentiate two we get zero and
don't forget we still have to integrate this one more time and we will end up with negative sine X
and here's the deal this times this is the first part of the answer we have here right here
and this times this is the second part of the answer which we have right here and check this out
I brought out this negative right here already
negative negative negative negative so we get the positive and then lastly this times this gives us
the last part of the answer which I put it down right here 2 times negative sine which is right
here but this is the positive because a lot of sign negative signs here negative times negative
that's why it's a positive this negative is from here right so this negative matches with that
so the answer for this I'll just put it down here so we have x square times sine X and then plus 2X
cosine X and then minus 2 sine X and of course don't forget the plus C otherwise everything's
wrong no you lose a point in my class I don't know if it's everything it's wrong in your class or not
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so believe in the diameter that's it