let's talk about integration by parts but i'm not going to talk about please let your students use
the di method because we did it already last year in this video i would like to talk about why i
don't use the LIATE method what's the LIATE method let me just give you guys example first suppose
we have to integrate x square times l and x and yes this right here needs integration by parts
but the hard part is how do we know which part to choose to be differentiated and the other
part to be integrated well that's what this right here comes in so the idea is that you
are going to go in this water if you see log functions in your integral you choose this
to be differentiated next is the inverse function such as inverse tangent inverse cosine etc and the
next is algebraic such as x to some power and then next is sine cosine for the trig functions
and lastly we have the exponential functions of course e to the x and things like that and as we
can see we see log right away so we'll choose that to be differentiated so let's go ahead and put
down a d and an i plus minus to get ready we will choose that and there's no thanks to that it's
pretty easy and of course we do x square to be integrated go ahead do that go ahead and do this
it seems pretty nice and let's just finish this and we'll talk so we get
one third x to the third power times l and x and then we are going to multiply by these
two things together and put that inside of the integral and subtract that here we have
one over x times one over three and then we have the x to the third power and let's put down the x
cancel this out and this right here becomes a two right and of course to integrate
this we just add one to the power which is 3 divided by the new power so we have 1 over 3.
so on all this right here is just 1 over 3 x to the third power l and x minus
1 3 times 1 3 so 1 over 9 and x to the third power done so this right here is very nice
and i can also give you guys another example so if you are looking at the integral of x times
sine x instead well we don't have the log anymore we have trig function but you see it's down here
so that means we should actually choose x to the first power because it's above the trig functions
so we'll choose the x to be differentiated so let's just go ahead and put that down put down the
x right here and put down the sine x right here okay x is algebraic x to some power and triggers
down below so do this and of course you all know the integration by parts with the dn method the
ds setup do it once and then do it right here but we can get to zero so let's do another row
alternate the sign let plus minus plus minus and also that
okay do this again we get zero do this we get negative sine x ladies and gentlemen
this times this this times this is the answer right so we get negative x cosine x
and then plus sine x dandy also put on the plus c right here again everybody happy so it seems like
this right here is actually a very good tip for us to remember and help us with integration by parts
but i uh first of all i will have to say i agree this right here helps about like
eighty percent of the time i would say the majority of the one of the classic calculus to
typical homework questions yes this right here will definitely be helpful but if you are trying
to do something like a hardwood version then that not really if you haven't learned all the
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number three let's look at the integral of secant to the third power x
we actually have to use the integration by parts yeah first off we are going to
break this apart into secant to the first power times secant to the second power so let's see
how it looks like so let me put this down right here first and then secant to the second power
x like that now we do have two parts and by the way do not replace this with a tangent
square plus one it's not going to help we have to use integration by parts right but anyway we
have secant to the first and then the other part secant to the second they are both trig functions
which partially choose to be differentiated right so this is one of the things that huh
yeah but anyway here is the video i really really like to tell my students that when you
do the integration by part with the dslr here always pick the part to be integrated first
because it's always harder to integrate something right well maybe that's not true
i actually just remember this not true but yeah anyway it's usually harder to integrate something
better than differentiate so just ask yourself can we integrate secant x yes by the easy no
can we integrate secant square x yes it's easy yes
so we will be integrating secant square x and let's just differentiate secant x
it works so integrating this we get tangent x and differentiating that we get secant x
another tangent x ladies and gentlemen this times this is the first part of the answer and we get
secant x times tangent x that's how we squeeze out the first part of the answer
for this integral next we multiply these two things together and then put that inside of
the integral it's a minus integral and we have secant and tangent to the second power
and this is the time that we will be using the identity and tangent square x is the same as
secant square x minus 1. so if you look at this carefully i'm going to actually
write this down again because you will see this is actually a repeating case
so we are looking at the integral of secant to the third power x dx that's equal to that secant x
tangent x this right here is instead of the integral but we'll have to distribute
the secant into here so we get minus integral of secant to the star power x and then dx dot
and we will also have to distribute this to that and negative negative so becomes the
positive here integral and then we just have secant x dx now you see
we can just move that to the other side and we'll end up with 2 integral of secant to the
third power x dx and you see this right here is just secant x times tangent x
and this is over there already and this right here it's a standard result and we just know
that that's going to be ln absolute value of secant x plus tangent x if you really want to
put secant x right here yes you can integrate that just by putting down a standard result
but once you multiply the second row the two things from the second row yeah no that's why
right always think about what to choose to be integrated first but anyway we are not done yet
because we don't want this 2 right here so divide everybody by 2 therefore the integral of secant
to the third power x dx is equal to one half secant x tangent x plus one half
ln absolute value of secant x plus tangent x and then of course plus c we are done
okay this is number four let's take a look at the integral of ln x over 1 plus
ln x and then square dx trust me we do integration by parts with this one all right so let's go ahead
and put on d and also the i plus one minus and thanks to this okay we see ln x which
is the natural log function and likewise we also have the natural log on the bottom
so it seems like we should just put the whole thing down to be differentiated ln x
over 1 plus l 1 x and square because both of them have a log right wow
do you really want to differentiate this it's doable but do we really have to is life that hard
you guys can let me know we really just have to think about which part we should choose to be
integrated first and we have to do it carefully so we just put down ln x right here to be integrated
no because we have to do integration parts for that right technically uh so the standing we
should just put on the rest and this is on the bottom so let me write down 1 over 1 plus l1 x
and then square can we integrate this no but what if we try to do use up let u equal the inside
well the derivative of the inside is 1 of x unfortunately there is no x
but now just think about it wouldn't it be nice if we happen to have an x right here
yes it would be so nice because then we can just proceed by u sub let u equal one plus l and x and
d u is just equal to one over x dx and you see this integral becomes the integral and then we
have one over u squared and this right here and the dx imagine that it's just going to be the du
and this right here is just going to give us you know negative 1 over u to the first power right
after you integrate that so this right here now is doable but we didn't have the x in the beginning
don't worry let's just go ahead and do the following if you don't have it multiply by that
and don't forget to do the same thing on the top
so now if we actually just do that you see we have the bottom portion right here that parts to be
integrated so what does that mean that means we should just take the top part
to be differentiated so let's go ahead and do that here we have the x times l and x
yes very nice check this out when we integrate this guy it's just negative one over u and u
is that so we end up with negative 1 over 1 plus l and x and when we differentiate this
guy we use the product rule so we do x times the derivative of the second plus the second function
times the derivative of the first which is just one just like that right and now let's construct
our answer like the go away right so we do this times this that will give us the first part of
the answer so we have negative and we have x l and x over that on the bottom and we are going to be
multiplying these two things together but check this out negative times negative is positive i
know i know no but let me just say that plus the integral check this out what's this this is just 1
and then plus ln x here we have 1 plus ln x when you will multiply this
they cancel each other out so in fact we just have a 1 right here and we have to differentiate that
right so this is so wonderful because this right here is just going to be plus 1. in the integral
so this right here is negative x ln x over 1 plus ln x and then we just need to add dx we are done
but i can make it slightly prettier for you guys because this right here actually simplifies really
well let's multiply the top and bottom by one plus l and x and then you see this right here
is just x plus x l and x when we combine this and that cancel so ladies and gentlemen this
right here actually gives us x on the top over that x over 1 plus l and x plus c we are done
so yeah this is what i like to show my students they just have to really think hard on which part
that they should be choosing from the integral to be integrated sometimes we just have to
make things happen and perhaps one of the best ways you can start by asking yourself
wouldn't it be nice and that's exactly what we did and that's exactly what happened
in fact i got this integral it's from the 100 derivative questions i i i have a question just
now how can i integrate this and get back to that that's a pretty nice question maybe
do some use substitution this is a hard one and this is what i come up with for you guys
so let me know what you guys think about the latte right here would you use it would
you not why or why not i would say it's pretty good about like 80 of the time but once we get
to like a trickier integrals then we really just have to think really carefully right
that's it