okay we have two integrals on the spot, the first one is the integral of x^ln(x)
second one is the integral of x^x we're all well unfortunately I don't have an
answer for this right here I do have a proposal by need to discuss with you guys so in this video
please just work on this first and then we will talk about this right here later on alright you
special function for that one pathway okay now here we go let's take a look at how we can solve
this well we have X to the X power already in this case we don't like to have the base being
X without our base e so let's do the following this is the same same as we all know we can try
XS e to the Ln X power this is still X however we still have that Ln X on the outside so that's what
we have in this case we see that we can just multiply the powers therefore we end up with
the integral of e and we have Ln X times the Ln X which is just Ln X and then square like that good
what can we do not oh well oh just try to stuff tends to happens right let u equal to Ln X in
this case we know X is equal to e to the U and perhaps that's just do it this way we see that
DX is equal to if the rope to this which is e to the u PU very nice and of course that's take this
thing to go to the you world and we see that here we have the integral of e and this right here is
the U and we have that to the second power and then DX is what just this right here which is e
to the U the U for the DX this and that up pretty good because they are both of base e so of course
we just add the exponents very nice and this right here reminds us the imaginary error function hmm
now will be great because as you can see this is almost at as soon as we complete the square
therefore let's look at this right here when we have use square plus you and of course that's
just to the magic number you take half of this and square that and then subtract it after that
which you end up with u square plus u half of 1 is just 1/2 and then squared is 1 over 4 so you
add 1 over 4 and then minus 1 over 4 up to that then the good news is this together is going to
give us a pretty square namely u plus 1/2 squared and then minus 1 over 4 isn't it yes so this is
all good to continue we see that this right here will keep us letting the grow up E and let's look
at this part which is u plus 1/2 of course I did in blue I should keep in blue and then we have
the TU well well this is the function part when we have e to the negative 1 over 4 that's just
a constant multiple so remember we can play this apart right so we can take that to the front so
when we do this times that we're going to this put up our work we will just have e to the negative 1
over 4 power all the way in the front it's just a constant and then the integral of this which is e
to the U plus 1/2 power and then square that to you and you can guess it of course in this case
we'll let T equal to u plus 1 over 2 so here let me just say T is equal to u plus 1 over 2 DT is
the same as the U so it's very nice there's nothing crazy we have to do so this is just
T squared right here and of course we can just use that finally you will see we still have this
constant multiple let's put it down which is e to the negative 1 over 4 power and then we have this
popping this which is a square root of pi over 4 ER f i and increase g which is just that so let's
put that down we have the u plus 1 over 2 but of course we have to go back to the X world excess
well you use Ln of X so right there for you and then we have the plus one over two and then over
perhaps that's good too and that together on the denominator so two and then this goes down the
denominator it becomes the fourth root of e very good don't forget the plus D of course who very
fun now I know this is the integral that a lot of you guys have been waiting for you guys have
been asking me for this but if you check work on alpha this not answer for it not even with
the elementary function so in that case what can we do well I've been thinking why don't we
start one ourself and then maybe we'll be able to come with some graphs or some derivative et
cetera on our own right so here is my proposal let me just write down this is just my proposal
I am not sure if this has been done what not but yeah it's just my proposal right I'm going to
define you see this is X to the X power there's a notation for that is that true with a little
X like this what little twin X like that but let me just define the following okay my proposal is
that that's call X I don't know about if you guys would like to have a better name just let me know
I use a capital X and then the little two meaning that we have two of these little X then two other
powers and in place X let's define this right let's define this to be the integral going from
zero to X this and that being the same but in that case the inside will be T to the T power
PEP that's kind of like how we come with two special functions because if you have a really
cool integral for example this one you can just use the error function and the error function is
defined in terms of integral right so why don't we call this integral to be started functions and
this is just between you and I okay it's just between you and I and from here you see I put
a little tube because we can do more because if you will like we can do like this X and then the
beat all three of X you guessed it yes integral going from zero to X and we could T to the T to
the T how crazy is that and the idea is that if we differentiate this function we'll just get X
to the X power if we differentiate this function we'll get X to the X to the X power very good and
remember we know how to do calculus in the sense that if we can get it derivative and the second
derivative we can get the graphs etc etc so maybe we'll do that later on I will show you guys how to
graph this dumb [ __ ] for now if you are ok with this that say the answer to this is X to write it
down this way x2 of X and we are done of course in this case put down Percy will all be happy what do
you guys think leave a comment down below if you have a better name leave a comment down below if
you are new leave a comment although I don't know yeah okay before we go I want to thank people in
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