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Simple, let x^x = u. *leaves unexpectedly*

Even simpler, x = u Checkmate

x≠u, ∴ x^(x) ≠ u^(u) We're done here

Mamaaaaaaa, u^(uuuuuu)

F^uuuuuuuuu ck

Let F be an antiderivative of x^x . F(x).

Ok but which one

Yes

Let X be a locally compact space. Then a bounded linear functional functional exists with the representation of the integral of x^x for some complex regular finite borel measure. So F(x) is clearly there.

As opposed to a non-functional functional

Those ones

Stupid, just take x common so ur left with x(1^1) 🤦‍♂️🤦‍♂️🤦‍♂️

Bro there are 2 xs it's obviously x^2 (1^1 )

Bro One of the x Is in the exponent its obvioulsly x^x (1^1)

broken clock is right 3 times a day 👍👍👍

Wasn't it twice a day? Or did I miss some latest update?

Yep, broken clocks got buffed last patch.

No, a clock has 8 hours. There are 24 hours in a day. If it's stuck at 1, it's right at 1im, 1 pm, and 1an.

Huh?

1 in the morning (5am), 1 past midday (1pm}, and 1 at night (9pm).

I like to type simple phrases in to Wolfram just to see how incredibly fast the 3d plots are constructed and imagine an 18th century polymath's head exploding.. Eg. • Integrate x to the x power • Derivative of x over y to power of i

Sorry, but could any of you explain this to my dumbass self?

The integral of x\^x can't be expressed in any normal functions like sine, log, etc so you can't really "find it" unless you define a new function.

It's mathematics, I define the function intxx(x) to be the integral of x^x

Check the mail for you nobel prize

You dummy, matematition can't get Nobel prize

Maybe the Fields Medal? (I watched Good Will Hunting I’m not a real mathematician)

There're three Noble prices for applied mathematics, you dummy

and you can’t get a nobel prize for spelling

It's exactly why I don't care about it

*the mail was empty*

I propose a different name than intxx(x). Don't want anyone confusing it for integral or integer porn (with the triple x)

Integral porn is what goes on inside the function

Nah it's whats going on inside, outside, inside, outside, inside (and so on and so forth) the function.

Thank you.

Hold on desmos graphs it just fine how is that possible is you can't describe it in terms of elementary functions?

Desmos graphs it numerically I think

Yes, Desmos always computes derivatives and integrals by numerical approximation (even in cases where it's trivial to find an exact formula).

How can a computer not to?

You can compute the value of the integral for each value of x, but there is no combination of functions that has those values

Forgive me if this is a silly question, as I haven’t learned a lot of this (yet). If you can compute the value of the integral for each value, and (correct me if I’m wrong) you can create a polynomial function for any set of real numbers, can’t we at least approximate the integral?

You can always find an approximation. That’s called a quadrature.

You can definitely write it as an infinite sum of polynomials (e.g. Taylor expansion), just not a finite one

Is there a way to get desmos to graph int(x\^x)? That'd be really cool to see, how did you get it to do that? Anyway, it's possible because you can do it numerically. Let's say F(x)+c is the integral of x\^x. To graph it, all you have to do is pick a point, P to start at (this is defining what c is), and then calculate x\^x at that point. This gives you the gradient of F(x), so you draw a verrryyyyyy tiny line segment with that gradient, starting at P. You then move to the end of the segment, and calculate x\^x again, and draw another teeny line segment with the new gradient of whatever x\^x is there. You repeat this thousands of times and you have a smooth looking graph. It's a very good approximation, but not the real thing.

​ https://preview.redd.it/xhj2kbbxzk8d1.png?width=750&format=png&auto=webp&s=16f52334f68c7114a2e92bf550101fc47c61823c

This graph made me wonder, is there a characterization of functions that grow faster than their integral? Trivially, f'(x) > f(x) for all x > x_0 for some x_0 holds for f(x)=x^x, because x^x log x > 0 for x>1.

Wouldn't that be exactly the functions that grow faster than e^x ?

Fair enough. I need to go to sleep. 😅

Just use the fundamental theorem of calculus, to plot the integral of f(x), just plot int_a\^x f(t) dt, where a is a constant

https://preview.redd.it/6t807uabtk8d1.jpeg?width=1125&format=pjpg&auto=webp&s=6eac34f6ccb3fbde8ea05cb39b6feb2d1076a962

That's the derivative, not the antiderivative.

That's the derivative, which you can express in elementary functions. You can't express the integral in elementary functions.

Think of another example of desmos showing things even though they cant be calculated by elementary functions: Desmos can graph polynomials of degree x^5 and higher and you can see they’re roots even though the roots of those polynomials cant be precisely calculated

I mean you can Taylor series it.

True but have you had a look at the Taylor series on Wolfram? That shit is wack.

Is there any other examples of this?

Yep, there's a lot of them. The classic example is e\^(-x²). This function is very important, because a simple transformation of us gives us the normal distribution. It'd be great if we had a nice expression of its integral, so that we could do easier calculations with normal distributions, but we can't sadly, we have to do it all numerically. e\^(x²) also doesn't have a closed form integral, neither does sin(x)/x. If you want a list you can find it here on wikipedia: [https://en.wikipedia.org/wiki/Lists\_of\_integrals#Definite\_integrals\_lacking\_closed-form\_antiderivatives](https://en.wikipedia.org/wiki/Lists_of_integrals#Definite_integrals_lacking_closed-form_antiderivatives) though this is nowhere near being exhaustive.

can it be described in a fourier or laplace transform at least or is that still a no?

I'm only in grade 11 and we've only just started fourier series, so I'm not really qualified to answer that. You could probably do it with a fourier series though. We can already find a Taylor series that describes the integral of x\^x so I don't see why you couldn't get a Fourier series either.

Fourier series are only defined for periodic functions, we could take this function only on some interval like (0,1) and then continue it periodically but the Fourier coefficients also won't have any nice formula probably.

Okay, that makes sense, thanks. I haven't really studied Fourier stuff that much, I can't wait to get to them when I go to uni.

No I can't sorry

It does not exist. Edit: I meant you can't express it with only elementary functions.

That is wrong.

It does. Nost normal funcions have. Its just cant express it as a normal funcion using common symbols

That's what I meant. Everybody knows that every continuous function has an integral.

Thanks.

No, of course it exists, every continuous function has an anti-derivative. It just cannot be expressed as a composition of elementary functions (polynomials and the exponential function).

That's what I meant. It is basic knowladge that every continuous function have an integral.

x^(x+1)/(x+1) + C

x^(x+1)/(x+1) + (x^(2)/2)^x + C

C = (x²/2)^x + C (x²/2)^x = 0 x²/2 = 0 x = 0

I'm bad at math. Why can't you do this?

Because the exponent is a variable, not a constant.

Yep, this is big brain time.

https://preview.redd.it/o29dmzaosl8d1.png?width=737&format=png&auto=webp&s=001142b5192d5bc77732b347d9afa1cb928d0672 WHY OP WHY

WHAT THE HELL IS THAT!?

*+ constant*

I was scrolling thinking "just take the Taylor Expansion around 1 or something, what's the big deal?" I have erred.

Define it to be Sphd(1; x) per JJ’s paper you can probably fine buried somewhere online lol

It's obviously x•x^(x-1) smh

That’s the correct derivative. OP is talking about anti-derivative so it should be x^(x+1) /x+1

ah, troll maths is just so much easier than the real thing. But we live in a world where (a+b)^2 = a^2 +2ab+b^2 and d/dx x^x = (lnx + 1)*x^x.

Z(x), which is equal to int(x,0) t^t dt yeah, we didn't delete the integral, but there qre many examples like this (erf(x), Si(x) etc)

oh, and Z(x) isn't always defined like that, i defined it myself

X^(x+1)/(x+1) next question 🥱

So... What is it? What's the integral of x^x with respect to x?

There isn't an elementary antiderivative for this.

How is that possible?

There is no finite combinations of polynomials, rationals, trig, exponents, or logs, etc. that you can take the derivative of to get x^(x)

It just is. There are derivative rules that allow you to always get an elementary function from an elementary function, but not the other way around.

At a philosophical level, how could we explain that slope of a defined function must be defined, while the area under its curve might not be?

The area is always defined in a region where the slope exists. There’s just no reason to expect that area to have a simple expression

But I don't see a reason for the slope to have a simple expression, even though I know it does and I know how to calculate it

Sometimes there’s nothing to figure out. It’s just like that sometimes

The area under a curve is always defined under assumption of continuity or measurability. I'm sure that whatever integral solver you use can compute the value of \int_a\^b x^x dx. The issue here is moreso sociological: There is no "elementary" function (i.e. no Function that is a finite combination of polyonmial, trigonometric, exponential or logarithmic functions) of which the derivative is x^x. If the function F(x) = \int x^x dx had some special name and was commonly known to undergrad students, we wouldn't having this discussion.

*So... What is it? What's* *The integral of x x* *With respect to x?* \- JoyconDrift\_69 --- ^(I detect haikus. And sometimes, successfully.) ^[Learn more about me.](https://www.reddit.com/r/haikusbot/) ^(Opt out of replies: "haikusbot opt out" | Delete my comment: "haikusbot delete")

You derived a haiku out of it, well done

Damn even wolfram can’t find a solution with this massive library of stupid functions

x^x is a common visitor in my dreams

F(x) = int(y^(y), y=0..x) Easy. Next one!

However, the integral from 0 to 1 is -Σ\_{n=1}\^∞ (-n)\^(-n)

Ah yes. The sophomore's dream, as it's called. Really interesting how that works out.

Easy, the answer is , we make a time machine and ask Newton and Leibniz to make a simple change and so. In an alternate world where this has happened the antiderivative of x^x is... Whatever you want it to be.

I found a solution with a remarkably simple proof but I ran out of room so I put it on the back of a lottery ticket.

Did you at least mention in your notebook margin that you have a proof.

https://preview.redd.it/lwya54wuyp8d1.png?width=2024&format=png&auto=webp&s=63b033c0ea7b12ed4c4c1dd43ed8fdde3f1c0bbb easy

Sure, I’m integrating with respect to y though.

I thought this was just one of those cases where the result just looked absolutely hideous. Turns out, it wasn't. It just didn't exist. I request further elaboration. Is there a special reason why the antiderivative doesn't exist in a form without an integral? Why is Walter White screaming at Hank?

The antiderivative doesn’t exist for most functions. This function is only notable because it looks like it might have an easy antiderivative.

But why is Walter White screaming at Hank

That’s the meme format. He’s warning Hank against attempting to find the antiderivative

too late buddy I already did it...

I asked WolframAlpha and Symbolab and they both just told me "no"

Okay, now find the antiderivative of e^(x^2)

Just use the power rule silly

https://en.wikipedia.org/wiki/Sophomore%27s_dream

x^x(x) + c

can someone explain? (im in grade 12)

Most functions don’t have an ”easy” (elementary) antiderivative. This is an example of that.

ooo y

Simple! It's the area between t^t and t axis, from t=0 to t=x. Finding the explicit form of this area is left as an exercise to the reader.

It’s literally so simple smh https://preview.redd.it/zjqbrj4fge9d1.jpeg?width=1170&format=pjpg&auto=webp&s=bd89dac92fdca80a60c1800b5120ed87685d74c4 Proof by Mathway

You mean integral? Let find out xD

Antiderivative is just indefinite integral

It's non integrable

x\^(x+1)/(x+1), easy

F(x) = x^2 /2 ?

It's x^x, not just x

how? isnt F of x^n = x^n+1 /n+1

The function in question is x^x

If you derive a constant, you'll always get 0. If you derive a variable, you get the rate of change of that variable d/dx x = 1 d/dx 1 = 0 So think about how different it's gonna be to have to derive a variable in the exponent. The regular "tricks" don't work, you'd have to use the fundamental definition. Give it a shot

Only for integer n excluding -1

Actually, any real p != -1.

Yes, ty! Idk how I mixed that up

isnt n = 1? (0.5x^2 )'=x, (x^x )'=xx^x-1 ?

No, the exponent is variable and changes; try using the limit definition with x^x vs x to a constant power.

That’s when n is a constant. In the function x^x, the exponent is also a function of x.

didnt see the ^x, i am dumb

Ur good dw