cauchy's repeated integral formula
integral and we have F of U 1 tu 1 and we also have the tu 2 like this
okay as we all know this right here seemed to grow but have you guys ever thought about what
if we have a repeating to go like this oh well in fact we have written s women for that and let
me show you guess right here so we will have to make some observation first right and the idea is
that we want to integrate the same function over and over and over so how we can make that happen
hey what do I have on my shirt right you see this is the D I set up for integration by parts and on
the I column you can just pick a function and you can just keep integrate that right so I will just
show you guys how we can actually use the t-i set up which is just the integration by parts
right to come up with stuff formula right so let me begin with two rows again I will just make the
observation which you get first and the idea is that I want to integrate a function so I'll just
put this down right here our coda function to be F and let me use the input to PT right and again
you can just now go ahead and just integrate integrate with how rows that you want but let
me - it's two for now okay integrating F I will get capital F of T and what I mean by this is
just integral let's pick some number a to t like this and then the little F function but for here
inside we cannot use that he anymore so you just have to change the dummy variable right because
you have to directly already so just use you right here like this so this is what I mean by capital F
of T you just integrate the previous part and you have the integral going from a to this right yep
that's pretty much the idea right okay now we have to keep in mind that whenever we are using the DFO
met when we multiply a row that's still going to keep us integral right so if we multiply this this
and that you see that will give us the integral of integral that's the type of integral that we want
very nice but I don't want to have anything right here to really bother me I don't want to have like
he wore like t square whatsoever signed him no none of that I just want to have a good number
so he behaves nicely this right here's naked he was so I would like to have a negative one as well
because the area times negative of course party and again integral of F which is the capital one
so its integral integral deserve good now we have to think backwards again I'm showing you guys how
we can construct the formula we want to end up with snake if you want so how can we get this so
derivative what would keep us naked you want in a tea world yes you're right it's naked et very nice
right so I will construct this right here for you guys first thing first we will multiply all this
and that's going to grow so we have the integral right and then I will just put down negative T
unfortunately we have two negative here but don't worry all right negative three times little F of
T and we are in the key world so of course I will just put down a DT right here that will
match with my integral and there people think red then we have to remember that when we multiply the
diagonals along with this time right here that's going to be the first part of the answer no more
integral in the front so let me put our negative feed right here and then well that's a batty I'm
sorry this right here negative T this is why I still can a whole stream or cross in one hand
all right and then we have to multiply by capital F of T so of course I'll read east on here for you
guys all right then we will have to multiply this row right here and that's you're going to give us
an integral nice enough negative times negative gives us a positive and the integral and that's
just one time stuff so just F of T like this and we have the the like that very nice now
we have the thing about this a little bit again this is what we want integral of an integral of
f so this is the top one integral we don't want this because this will give me that right if we
don't have this right here so now we just have the thing about maybe there are some numbers was
an expression so we can plug in so that we can actually carry off this part all right so just
keep that in mind let's say this interval goes from tea from some number to something else and
in that case you would be the same right here and that case it would be the same right here as well
so what are the numbers that we can plug in and I want make our life much easier a what worked out
nicely I'll put down down right here write this a why because if when I put on a right here capital
F of a capital F of a is just integral going from a to a of the little F which is going to
give me zero that would be really nice so we don't need to worry about this part right so I'm gonna
start with all this T goes from a that's good now what should I put on right here though for the T
on the upper limit so I put on zero if I products is running to that he okay that'll cure this term
that's good but the problem is that if we just go from a to zero this is just going to give me
a number the post I want to integrate a function of course we know that at the F we should also end
up with a function plus a constant so we cannot really just put on zero for the upper limit so
what can we do hmm well first of all remember we want to end up with a function so perhaps we
just really have to have a variable right here and let me just use X right here and when I put on X
this will also have to be X now we also have to be X right so that's good but I promise that if
I put an X right here this doesn't give me 0 it's negative x times capital f of X no good but can we
stop changing the third little bit yes because remember I just need to end up with negative 1
right here so we had to ask ourselves derivative what in the tea world will give me negative 1 well
earlier with that it's negative T but in the T where X is a constant so can we actually put
down X minus T right here yes so again remember we're constructing a formula so this is exactly
what we're trying to do all right now if we have the X minus T there are just how to come back
here and fix this a little bit because this right here multiplied the first roll I will just have X
minus C times little F of T and that would be in self that you think the grocer just had to
fix that and then I would do this times that I was to fix this part and this part states the
same because I still end up with this this and that multiplying together in something they grow
very nice and now check this out when I put the X right here plus X minus X yes it's 0 so this term
based on all this you will be 0 very good huh now something serious let's write down the left-hand
side and I'll try to write down this from a to X and I'll try to keep although actually I'll just
write it down this is X minus T times F of T and again we are in the T world equals again this part
what is just your role and now this right here is the integral going from a to X and we have
the capital f of T which is just the integral of the little F right but we'll just have to go from
little A to D 2 T so it go from a to T dito F and increase you do you but on the outside we
have the T and that's exactly how it just did we have a double integral integral intergroup
these are f on the right hand side if you look at this from right to left we end up with a
much simpler what should I say much simpler on life it's much simpler or not but instead of
two integrals that you have to do you actually have to just do one and that's very nice right
so this right here it's a good idea now let's see if we can extend the idea to more integrals
we'll be doing the same thing but this time between full rows for you guys and hopefully
we can see a nice pattern after this so here's the plus minus plus minus we will begin with
integrating e to F of T well I will just have capital F of T and this means integrating from
a to T of the previous which is just F of and you have to use a different variable for that how to
say you that we have more to go so I'll say you want for the dummy variable like this now we have
to integrate capital F of T that will give us F of F of T like this capital F just means that we
are going to integrate the previous so this means I will have to integrate the previous so just put
this down right here which is the integral and I will just write down F of U 1 D u1 but we have
to be really careful with this because I cannot just put down a 2t because that he has to be a
berry on the very outside right here because the output here should be a function of T right so I
will actually go from a to a not a variable let's say you - and then we'll be in the YouTube world
so that means we don t you - right here and we'll be going from a to T right here for the outside so
we end our itself on shaking terms of T like that don't worry what do you get for including this one
it's F of F of F of T like this yes three of them and this right here it's going to be integrating
this right so we will have to just read this times I which is the integral integral and we have F of
you won t you want and we also have the t you - like this and we see that for the very inside
we go from a to you - and you picked it up in the YouTube world so in the YouTube world we again go
from a but we cannot go to t because you have to say for that he for the outside right here
right for the outer one so I will just choose you Street right here and then we have the you street
like this and then we were just going from a to T on the very outside like this so this is what
F of F of F of T means right so it's just this part right here all right now this right here is
again negative so I want to end up with a negative one yes right well the rupee of what will give me
negative one in the tea world yes just the earlier we need negative T but per our discussion earlier
right when you want to have X minus T so that's pretty much the idea we just have to now go back
so we will see a very nice thing this time we have to integrate right because we're going backwards
pretty much yeah how many integrals were doing well just pretty much integrate this with respect
to T you pretty much will have this again namely X minus T by you have the second power you just add
one to that power and divided by the new power which is divided by two but we have to use the
dual chain because the derivative of X minus T in the T world is negative one so we have two divided
by negative one as well so let me just multiply by negative like that all right so we have negative
one over two of all that don't worry what do you get pretty much just integrate again rise of the
function power pretty much stays because yeah it's linear inside and we have X minus T to the third
power because we add 1 to the power and then we divide it by the new power so for the denominator
we will have three times two very nice huh and then negative divided by negative will just give
us positive one very nice and you can see that this is 3 times 2 yes if you would like and also
put on times 1 likewise this is 2 times 1 thanking imagine if you have to keep going up you have 1 to
the power and divide by the new power you have 4 times 3 times 2 times 1 namely 4 factorial this
is 3 factorial this is 2 factorial and then one veteran and also no factorial what's your factor
technically all right so here is 30 oh very cool stuff remember this times this this times this and
this times this they are the first pop the answer to have that integral right stop originally this
is the integral we have integral this times that this is the very cool stuff right so from this we
have the integral and per our discussion earlier we actually have to go from T equals a to T equals
x ok just keep thank Matt so now I just have to put this times that in the air in the T world good
and then remember we have to do this times this this times this this times this along with the
sign in front and I'm just going to write it down right here for you guys and now remember we have
to put all this and we have to go from a to X and this right here you just have to plug in and walk
you out right lastly we have to hook this times this times this in self integral you see this
right here it's the trip we took for like this but anyway let me put this down first though we will
have to add right because negatives negative so this lastly we have to integral T goes from a to
X and how it's just put this down right here for you guys so we have the capital f of capital f of
capital f of t like this and on the very outside we have the T world right so it's like this right
here and again it's pretty much three integrals in self this dragging across we are all together for
fit very cool stuff and now here's the beauty when you put X into this tower p 0 likewise you wrote
this right here also 0 then here's the deal when we put a in here capital F of a well it's just
the integral going from a to a so that will be 0 and when we put little a into this right here
capital F of capital F of a it's just this part you see if we put the a right here it's just going
to be the integral going from a to a and again that will give me 0 as well similarly for this
one F of F of F of a that will be 0 as well in other word everything right here will be 0 very
nice huh and as you can see right here we have the integral and remember this way it was three
more zeros right here so I'll need right-hand side we actually flowing the growth and you look at the
left-hand side we only have one to do so we can go from falling the growth right here to just 1
by doing the integral from a to X and we have 1 over 3 factorial times X minus to the third
power and F of T TT and notice that when we have four integrals this is G factor Odysseus Street
what if we have an intercourse then well would be just 1 over n minus 1 factorial times X minus T to
the N minus 1 power times F of T T were and just going to greater from a to X that's it
in fact if you would like you can extend the idea to not just pass the whole number n but also maybe
like 1/2 right man I'm going to work out for you guys next introducing half integral for your
guests coming soon right bye anyway this is it for now I just want to show you cast a formula and
that's how you can derive it just think about this backwards and hopefully guests all like this video
thank you guys so much for watching and as always that's it