okay here we have the integral going from 0 to 2 of x to the t-th power dt and you might be
thinking that we're just going to integrate this right here no we are going to say this
right here is equal to 3 and we are going to find out for what x will make this equation true
and let me tell you this is going to be a very very hard equation because not only we have to use
calculus but also the lambert w function with some technical parts and of course most importantly I
came up with this and I'm pretty sure 99% of the world population wouldn't know how to solve this
but anyway let's see how we can deal with this first off we will have to know that
this integral is in the t world and the x is just like a constant so we can actually just integrate
this no problem on that and let me just remind you guys on the side when we integrate let's
say 17 which is just a constant raised to the t power dt this right here will give us
17 to the test power over ln of the base 17. so the exponential function when you integrate that
you keep it but you divided by ln of the base so in this case, we'll just have x to the t power
divided by l and x then that's the calculus part and then, of course, we'll just have
to plug in plug in zero to two and then make this equal to three so that's what we have now
that's the calculus part remember we are in the t world so we will have to plug in the 2 into the t
so first we get x to the second power and then minus plugging 0 here and x to the zeroth power
and both of them have the l and x so i'm just going to put that down right here like this
and this is equal to three okay and then perhaps we can write this down x plus zero's power as
one and maybe we can multiply the ln x on both sides because usually we don't like to see
fractions in the equation so we are getting x squared minus 1 being equal to 3 l and x
now how in the world we solve this kind of equations
right
newton's method no no no no no i know it works but let's try the lambert method
with the number w function and here's the idea like how do we know to use the number w function
when we have a power function x to some power and then natural log where exponential e to
the however sometimes the lambda w function might be able to help us out so just keep
that in mind i'm not saying all the time though just keep that in mind
and i would like to write down the most important note for you guys when we use the
number w function i'll put it down right here and i know you guys like me to draw the fish
so i will give you guys a fish suppose we have a fish in the front this is not alpha
this is a fish it's a happy face times e like the special constant e raised to the same fish
all right if this entire the same and once we apply the number w function guess what we will
just get the fish back and this right here is the most important part of using the number w function
so with that being said
we kind of want to introduce the e to help us out a little bit don't you think so all right so what
we can do is let's just do e to this power and e to the power all right cool on the left hand
side we have x squared minus 1 in the power let's break them apart so we will get e to the x squared
times e to the negative 1. that's legit and then for the right hand side we have the e and ln but
we cannot cancel it out yet we will have to first put the 3 back here for the to become the power
and then cancel the e and then ln so the right hand side gives us x to the third power
all right so now we are working with power and also exponential all right and you see
we kind of want to have the fusion finish on the same side i want to have the function part
being on the same side so let me divide the x to the third power on both sides
and let's kick the number to the other side and this is what we are going to get
divide this to the other side we will get x to the negative 3 power and then we have e to the
x squared power and then we divide this on both sides 1 over e to the negative 1 is e to the first
power so i'm just going to write this down as just e cool i think we are making a good progress
notice here we have x squared but this is x to the negative three we don't want a negative three
we want to have this right here being 2 as well how can we make that happen
though well this is the power so what we can do is just go ahead raise both sides to the
negative 2 over 3 power and of course we do it to both sides because this way we can distribute
the power and inside just a multiplication so we will just get x negative 3 times that
we get positive 2. very nice huh and then when we distribute the power we do x squared times that
so we get e to the negative 2 over 3 x square power and this right here gives us
e to the negative 2 over 3. i know it looks pretty crazy now yeah if they
have it has been crazy like maybe right here already all right that's pretty good but now
we need to have the negative two over three we're almost there so what can we do yeah just multiply
that on both sides we need a negative two over three let's multiply negative two over three on
both sides and then ladies and gentlemen here we have the fish and here we have the same exact dish
therefore we can just go ahead and take the happy number w function on both sides like this
cool and now check this out on the left hand side this and that are the same so just like that we
know we will just get negative 2 over 3 x squared done on the right hand side you might notice this
is very interesting because this and are the same so by this again hey we should just get negative
2 over 3 right here as well right and guess what let's continue divide this on both sides we get x
squared is equal to 1 and do the usual business we are saying x shall be plus or minus one
no this is not correct this is not correct why if we put one back to the original equation this is
an equation which is the one here one to the teas power is just always equal to one right in the
real world so this right here does not work and now you might be wondering how did this process
give us one for the answer though well i'll tell you it's because it's right here
when we had x squared minus 1 is equal to 3 times ln x if you look at just this equation
right nothing else then let me tell you one is indeed a legitimate answer to this equation
because you will get 0 on the left-hand side and also 0 on the right-hand side but
you always always have to refer back to the original equation to see what we are trying to do
and then get rid of the one in fact one would then apply once we get to here because we get
l and one which is zero on the bottom that doesn't work either and then negative 1 is
just outrageous so let's not even talk about that so what exactly did this go wrong though
did the number w function fail because where's my answer
right so hopefully by now i guess you'll see why this is hard and of course i have
the answer for you guys don't worry i would like to tell you guys this on the left hand side it's
okay i will just write it as negative 2 over 3 x squared that's okay on the right hand side
i would like to tell you guys with some more notes on the side right here earlier
when we just look at the lambda w function of negative 2 over 3 e to the negative 2 over 3 power
this right here will indeed give you negative 2 over 3 because this and now the same good but this
is the so-called principle value or the principal branch it's kind of like saying the first answer
and to make the principle branch clear usually we say w with a subscription rho here all right and
that's why they use a word from alpha as well but unfortunately the principle value does not help us
to solve this equation so what do we do then just go to another branch another branch
and let me tell you we will actually still end up with another real solution once we get to
another branch so what's the other branch i wouldn't tell you it's the negative 1 branch
and we get negative 2 over 3 e to the negative 2 over 3. you can use both alpha for this and i
have the answer written down over there already this right here is approximately negative 1.429
so this is the answer that we are going to use right here
all right and now you might be wondering why why does this give you two answers let me tell you
pay close attention to the input this number if it's in between of well let me write it down
this number which is negative 2 over 3 e to the negative 2 over 3.
if this is in between of negative 1 over e and 0
then we are going to have two answers right we are going to have two answers
and let me just tell you guys that this right here this number right here is approximately
zero points three four two just kidding it's negative and this number right here negative
one over e is negative zero point three six eight all right so that's why you see this is indeed
between this of that that's why it will give you two answers when we use the number w function
and you might be wondering are there any more answers yes if you go to the other branches such
as the negative 2 branch or such as the first branch you end up with a lot more answers and
in fact they are all complex answers so yeah it's really crazy huh i would like to keep the answer
in the exact form so let me just write it as w with a subsequent negative 1 and then you really
have to write this down as how it is which is negative 2 over 3 e to the negative 2 over 3 like
so and finally let's just do everything in one go i'm going to multiply the reciprocal on both sides
and then take the square roots on both sides so x is equal to we will have that big square root
and we should have the plus or minus whenever we take the square roots on both sides
but we are staying in the real world we cannot have the negative for the base for the exponential
so x will just be positive so i'll just erase that okay and then once we bring this to the
other side it will be negative three over two and then this right here is the number w
function negative one branch and then negative two over three wow this is just so beautiful
negative two over three here and yeah so ladies and gentlemen this right here is the answer
what am i talking about right it's just a bunch of letters and like numbers some symbols how big is
exactly this i would like to tell you this right here uh it's approximately it's approximately this
right here x is approximately one point six one point four six four so on so on all right
and you might not be comfortable why we have a negative self the square root but remember the
blue part is actually a negative so negative times negative we get the positive so we are all good
yeah let me know what you guys think about this right here for more videos like this one i have
another one for you guys over here and also check out the video in the description yep that's it