how WolframAlpha define these non elementary integrals
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- Опубликовано: 16 окт 2024
- 8 special functions for your non-elementary integration needs, as seen on WolframAlpha!
Shop non-elementary t-shirts: blackpenredpen...
1. Error function erf(x) en.wikipedia.o...
mathworld.wolfr...
2. Imaginary error function erfi(x) mathworld.wolfr...
3. Exponential integral function Ei(x) en.wikipedia.o... mathworld.wolfr...
4. Logarithmic integral function li(x) very important in number theory/prime numbers
en.wikipedia.o...
mathworld.wolfr...
5. Sine and Cosine integrals Si(x) and Ci(x) en.wikipedia.o...
6. Fresnel's integral S(x) and C(x), en.wikipedia.o... mathworld.wolfr...
@blackpenredpen
Now you can integrate e^(e^x) and ln(ln(x))
Think about them first before you watch the solution here: ruclips.net/video/tDEjCci8lrk/видео.html
Math for Fun,
Functions for Integrals,
Integrals for Pleasure,
YAY!!!
BPRP. What else?
What will be the integral of x/e^x ___>>
@ Write it as x * e^(-x) and use integration by parts
can we use series expansions to get these special non-elementary functions?
Since they are both non-elementary, we can get this special function as this answer. The integral of e^(e^x) is Ei(e^x)+C. The integral of ln(ln(x)) is xln(ln(x))-li(x)+C.
The mathematician(s) who invented these notations be like:
“If I can’t integrate it, no one can!”
You can integrate them, you just can't write them as an elementary function. So you have to either do this or write out its Taylor series.
Andrew Blechinger
I’m well aware of that I’m just messing around 😅
Thank you for being yourself! You make math so enjoyable and sincere!!! You have been my favourite math youtuber for 3 years. I've been following you since 11th grade and now I'm a sophomore. :)
Always making us challenged with newer integrals. I'd love to take your lessons in person. Love from Turkey!
*2 ! ❣️
@@emiliomontes2043 Why 2? I'm following him since this video of his: ruclips.net/video/SPHD7zmLVa8/видео.html
(high school 11th, high school 12th, uni 1st and now uni 2nd) Guess your high school is 3 years :)
Thank you so much Evren! A comment like yours always makes my day!!
@@blackpenredpen ❤
@@evreng He meant x2. Meaning he likes BPRP too
49% of the comment section: math
Another 49% of the comment section: physics
2% of the comment section: wow I dig your haircut😍😍
0.00000001% of the comment section: 3:05 wow that voice crack tho 😂😂
Carlos Teveth hahahha nice catch!
nobody mentions that nice new hair cut? so i'm first!
switchhax awwww thank you!!!!!!!
@@blackpenredpen reminds me that i need to also go get a new one 😅
2nd
For one who doesn’t know much of math : number 5 is Si(x)+C =6+C
So 5 = 6 + C, meaning C = -1. It all makes sense.
I think of the bell curve intelligent meme.
@@choiyatlam2552[insert relevant error function joke here]
I'm a surveyor. The Fresnel integrals together make the Euler spiral (aka Cornu spiral or clothoid), which is used in designing highways and parkways. My program Bezitopo uses spirals to draw contour lines. Turn the error function 90° and you get the imaginary error function; turn it 45° and you get the spiral.
Fun fact: the Euler spiral is also used for vertical loops on roller coasters! They’re often referred to as clothoid loops.
was feeling down but these special functions have cheered me up thanks ! 🥰👍🏾
halalxan glad to hear! Is everything ok?
@@blackpenredpen just unable to find a job for the past 4 months sadly. but applying everyday staying hopeful 😁
halalxan
Best of luck to you. I understand how frustrating and depressing it can be because I had a similar situation before. I use simplyhired to search for jobs and maybe you can try that if you haven’t used that before. Also have you consider to do some tutoring on the side? That can help with a bit. And if you have questions on that, let me know and maybe I can give you some ideas.
@@blackpenredpen i actually applied for a tutoring position the other day for children aged 4-14 and managed to get through for an interview and practical assessments. I didnt do the best with the interview and was actually surprisingly pretty awkward working with the kids hence didnt get the job 😓😓. i will look into the simply hired website thanks!👍🏾
halalxan I see. Best of luck to you! Keep us updated!
I'm a physicist. The Fresnel integrals S(x) and C(x) are used in the near-field approximation of optics, which is not my specialisation. However, I don't remember having used si(x) nor ci(x).
We don't usually work in terms of transcendental integrals, but rather in terms of series solutions to differential equations: Airy and Bessel functions, Legendre and other orthogonal polynomials, etc.
As a basic example, the exact period of a pendulum can be given in terms of an elliptic integral of the first kind, but it is commonly expressed as a series expansion in terms of Legendre polynomials.
Fun fact: the exact period of a pendulum can also be written in terms of the so-called arithmetic-geometric mean agm(x,y). Start with two numbers x and y and calculate both their arithmetic mean and their geometric mean, and you will get two values. Then take the arithmetic and the geometric mean of these two values and get two new values, then again and again at infinitum. The two values obtained from these iterations of arithmetic and geometric means will quickly converge to a single value, the arithmetic-geometric mean of x and y, agm(x,y).
The exact frequency of a pendulum is (1/2π)*√(g/L)*agm(1, cos²(θ/2)) where g is the gravitational acceleration, L is the lenght of the pendulum and θ is the initial (maximal) angle of the pendulum. Notice that for very small θ, cos²(θ/2) is approximately 1, and agm(1,1)=1 and so the period reduces to the familiar formula.
Dani Borrajo Gutiérrez si(x) and ci(x) are more commonly found in engineering. However, si(x) is one of many solutions of the Schrödinger equation for a specific potential.
They're both used in quantum mechanics I'm surprised you haven't come across them before.
Sin(x^2) and cos(x^2) integrals are Fresnel integrals used in optics light diffraction explain.
Phyarth Explain what?
@@angelmendez-rivera351, diffraction or interference intensity magnitude is proportional to square law.
www.thefouriertransform.com/applications/diffraction3.php
@@phyarth8082 Explain what?
@@pbj4184 Such mathematical spice in this video is used for Fresnel diffraction meal :) en.wikipedia.org/wiki/Fresnel_diffraction
@@phyarth8082 Can you speak English? You said to explain. Explain fucking what?
The world does not deserve how great this man is!!!
The exponential integral function Ei( ) can also be used for question (4):
∫ 1/ln(𝑥) d𝑥
= ∫ 𝑒^(ln 𝑥)/ 𝑥 ln(𝑥) d𝑥
= ∫ 𝑒^(ln 𝑥)/ ln(𝑥) d(ln(𝑥))
= Ei(ln(𝑥)) + 𝐶
Wow how to write that
He does the exact same in the video
100 differentiation
100 integrals
100 series
Every SINGLE day
ONE INTEGRATE MAN
One calculus man sounds Better ngl
Great video. My mentor in college thesis advisor’s thesis advisor was G.H. Hardy and he did analytic number theory. I think you would like analytic number theory. It has a lot of integrals especially integrals in complex analysis. Really, you should check that subject out.
For many integrals I've put into WoflramAlpha, I've been given that the answer involves an "elliptic integral of the second kind", or something similar. Would you be able to explain what that means?
I will dig into that more later. I will give some exercise with these special functions first, then Li_2 and elliptic integrals.
@@blackpenredpen Thank you! I've attempted to read into it myself but the information goes entirely over my head.
@@blackpenredpen cool!
armin The incomplete elliptic integral of the second kind is a function E(φ, k), and it is exactly equal to the integral from t = 0 to t = φ of sqrt(1 - k^2·sin(t)^2). It as simple as this. The complete elliptic integral of the second kind is equal to E(π/2, k) = E(k).
I always wanted a nice summary of those special functions!
Also, nice haircut :)
Trying to get the hang of these hopefully in future I’ll be flying through them,thanks for everything🙋🏿♂️
I feel like Marlin in Finding Nemo: “It’s like he’s trying to tell me something...”
Second part using Li²(x), Ti²(x), F(x|m) y E(x|m)?
Thanks!!!
can you do some non-elementary functions
I really like this
Hey blackpenredpen ! Do you know how to prove properly that those functions don’t have any anti derivative that can be expressed in term of elementary functions ?
Greetings from France
Anthony
Antho Risch algorithm. See Wikipedia.
Just giving a shout out to the elliptic integrals that were not mentioned in this amazing video.
These 8 integrals are as cool as yr new haircut 😎 Thank you man!
Mak Vinci thanks!!!
Atomisation is a one of the best way of derivative or integrals. Switching f(f(f()))).. 7&8 squared equals one.
Great. Please make something about eliptic integrals. Best regards
Lots of incomplete Gamma function related stuff. So for example Gamma(1/2,x^2) relates to erf(x).
Really amazing "natural" function en.wikipedia.org/wiki/Incomplete_gamma_function
Actually, the first time I’ve seen the Fresnel integrals was when I started to create my proof that Apery’s constant cannot be expressed in terms of elementary functions. That also means that Apery’s constant is transcendental and that (Apery’s constant)/(pi^3) is transcendental since both Fresnel integrals are transcendental functions and the sum of them don’t result in an algebraic function. I’ll send you an email about my proof when I have the time. ; )
Sincerely,
Alex Eduardo Gonzalez
I wish i understood a word of what you said xD
Metalhammer1993 Ok, then I can explain what I mean. This is a long explanation, but it will be worth reading, as I explain it in the simplest way possible. Apery’s constant is a constant defined as zeta(3), where zeta(s) is the Riemann zeta function, which is defined to be the summation of (1/(n^s)) for a given value s, which is usually an integer when talking about the real world, that is, when the imaginary part of a complex number is 0. The reason it’s called Apery’s constant is because it was named after French mathematician Roger Apéry, who proved that this constant is irrational. Near the end of my junior year in high school, I used Steve Chow’s (aka blackpenredpen’s) definition of the Fourier Series and then Parseval’s theorem, since there are different definitions for them, to prove that it cannot be expressed as some elementary functions, which are functions that are taught from elementary school to high school. You may encounter some of these functions in college that were from high school, but you’ll encounter new functions along the way. Those new functions taught at college and beyond are non-elementary functions, such as the error function and, yes, the Fresnel integrals. I have a proof, but for the sake of simplicity, I cannot write more in this reply, or that might count as spamming and I’ll get a Community Guidelines strike. Thus, I will release the proof on Saturday, September 7th, 2019. By proving that Apery’s constant contains non-elementary functions, it’s also transcendental since we’re not dealing with algebraic functions, but transcendental functions. Hope that helps! : ) By the way, if you want the proof now, you can ask me. Thank you for reading!
@@einsteingonzalez4336 thank you for the explanation this really is interesting. i think i can wait the time, were/how will you release it? just so i can stay tuned? gotta show it to my old maths professor. That sounds exactly like something he´d be interested in. ( i studied chemistry. So i know what the zeta function is and what a transcendental number is, but i never had to deal with transcendental functions (and transcendental numbers were pretty much a throwaway line. like "pi and e are transcendental, but don´t bother dealing with it") but i´ll definitely watch out for your proof.
Metalhammer1993 Thanks! Also, I created this in Microsoft Word, so I’ll try to publish the proof in a PDF file or convert my paper into a Google Doc. Unfortunately, life has blocked the way for me, because I have my 3 nephews, 1 niece, 1 sister, and 1 brother to take care of and at the time in the making of this reply, my mom is playing Solitaire on his iPhone and on my bed. Also, while my paper contains the proof that zeta(3) is transcendental, it also goes to all of zeta(2n+1). In other words, the goal in this paper is to prove that zeta(2n+1) for all positive n is transcendental, which includes zeta(3). It’s not zeta(2n-1) because if I plug in n=1, I get zeta(1), which is infinity. I already showed my proof to my Pre-Calculus teacher, Mr. Krolikowski, so technically, I can show you a picture of his whiteboard with my proof for zeta(3) only in there. Right now, however, it’s nighttime. See you tomorrow. : )
@@Metalhammer1993 Ok, here it is! Just by reading the link, you can be sure that it leads to a Google Doc, but I can also assure you that this leads to the proof. I had a rough week of school, so I have to share the link to satisfy most, if not all, of your expectations. Enjoy! docs.google.com/document/d/101txhBfBCGHNfjD0Sfd6Z1T-nqrgpcwiQGJuRE2plwo/edit
The integral of sin(ax)/x dx from 0 to ∞ is equal to π/2 ONLY when a is positive. For negative a the result is -π/2 (since sine is an odd function)
Yea. I fixed that later in the video.
Could you discuss Risch algorithm? I would like to know how to verify if an antiderivative of a given function is an elementary function or not. Besides, some special functions can be finally included into the set of algebraic formulas regarded as "elementary" - perhaps ln x is a very good example. So, does the use of such functions as ln x or erf x allow to express antiderivatives in some finite algebraic formulas using special functions as their components ? Another issue is integrating inverse functions. Is there a general method of integrating inverse functions ? It would be interesting, for example, to integrate ssrt (x), although I'm afraid it's also non-elementary...
Excellent, very good information. 😄
There was only one time I used one of them. Then I was thinking of different easier integrals to solve, I've come up with the integral of "sin(1/x)". At the end, as an answer I got "sin(1/x)x - Ci(1/x)".
“The test only has 8 integrals”
The integrals in question:
Next video: write some of these functions in term of the other using subtitutions and complex analysis
Good evening from Poland. ;)
I've been battling an integral that im pretty sure is non-elementary because all the normal methods just don't work and it contains parts that are non-elementary on their own
X --> Steve hairs
Lim(X)
X---->0
Amazing, just amazing
Is there any integral like those whose result is pi/sqrt(2) :') it's missing
Infinite What are you talking about?
7 and 8 can be used to paramerize the spiral
Sir make video on advance mathamatics like partial differteation or doubel integration
thanks
I have a special integral for you where WolframAlpha says that it couldn"t be done but believe me this integral is elementary:
I = int of (e^x-1)*dx/[(e^x+1)*sqrt(e^x+e^-x+1)].
This is like saying water is water.
Correct me if I'm wrong, e^(-x)^(2 ) we can multiply -x and 2 by exponent properties and then it becomes e^(-2x) and it is very easy to integrate ans=(-1/2 e^-2x) is it correct...??
no, (e^x)^2 menas you multiply (e^x)*(e^x)=e^(2x), whereas e^x^2 means that you only sqare the exponent of e-> e^(x^x), only (e^x)^x would be the same like e^x^x
Make some videos on metrics chapters. And vector chapters
Do you get invites to functions, to talk about functions?
You are a genius
Is there any special relation between (7), (8) and (1) and/or (2)?
If you add (8) + i(7) thanks to euler's formula you would get the integral of e^(ix²) dx which is close but not quite there.
7 and 8 are both Fresnel integrals. 1 and 2 are both related to each other since they are error functions. One of them is real and the other is imaginary. erfi(x) stands for the imaginary error function. erf(x) is an error function.
@@justabunga1 I don't mean a connection between 1 and 2. I mean a connection between the pairs of 7-8, and 1-2.
mrBorkD Yes, there is.
@@angelmendez-rivera351 and it is...?
All 8 integrals can be written as special cases of the incomplete gamma function.
An awesome video! But how can you miss out on the famous integral of \sqrt{x^3+1}
yingqi Wen This integral is not exactly famous, though. There is no function defined with respect to a multiple of this integral.
What about erfc(x)? or is that just another name for erfi(x)?
If for the first function in the above examples, what will be the answer if we take integral from 0 to 2pi
I love ur smile Sir 😘
What level are these taught. I took Multi Variable and DifEq and never learned these integrals.
the integral of ln(arctanx) dx pls!!!!!we can witness new milestones in Calculus!
francis kyle flores u = ln[arctan(x)] => x = tan(e^u) & dx = sec(e^u)^2·e^u du. Therefore, ln[arctan(x)] dx = sec(e^u)^2·ue^u du. If one integrates by parts by differentiating u, then one needs the antiderivative of sec(e^u)^2·e^u with respect to u. Letting e^u = t gives that sec(e^u)^2·e^u du = sec(t)^2 dt. The antiderivative is simply tan(t) = tan(e^u). Therefore, integrating by parts gives u·tan(e^u) - I2(u), where I2(u) is the antiderivative of tan(e^u). Substitute and you get an answer. The antiderivative of tan(e^u) cannot be expressed in terms of elementary or special functions.
Is sec^2(x) -tan^2(x) a periodic function or a constant function?
vishal sharma It is both, since sec(x)^2 - tan(x)^2 = 1, and 1 is a constant function, and all constant functions are, by definition, periodic.
Can you integrate 1/[arccos(x•sqrt(1-x^2))•ln(1+sin(2x•sqrt(1-x^2))/pi)] dx
The black dot is for multiplication. I failed to attempt it sir.
Why do you need to solve such a difficult integral? Have you tried plugging it into WolframAlpha? If you only need an approximate solution for small x, try using a power series
My calc teacher uses a calculus exercise book writen by a russian whose surname is Demidovich. One of the problems in the book requiseres the use of the fundamental theorem of calculus 1. It asks to take the derivative of F(x)=integral from 0 to x of sqrt(1+t⁴)dt.
Pretty easy beacuse you only need to use the FToC. But, what if the problems was to solve this integral? Neither Wolfram Alpha nor other integral calculator was able to solve the indefinite integral of sqrt(1+x⁴)dx. This isn't elementary. But there os a kind of function that we can use to force a result for this integral?
Victor Paes Plinio Integrate by parts, and choose to differentiate sqrt(1 + t^4) and antidifferentiate 1. The derivative of sqrt(1 + t^4) is 2t^3/sqrt(1 + t^4), and the principal antiderivative of 1 is t. Thus, the integral from t = 0 to t = x of sqrt(1 + t^4) is t·sqrt(1 + t^4), evaluated from t = 0 to t = x, minus the integral from t = 0 to t = x of 2t^4/sqrt(1 + t^4). t·sqrt(1 + t^4) evaluated from t = 0 to = x is simply x·sqrt(1 + x^4). 0 = 2 - 2, so 2t^4 = 2t^4 + 0 = 2 + 2t^4 - 2 = 2(1 + t^4) - 2. Hence, the integral from t = 0 to t = x of 2t^4/sqrt(1 + t^4) is equal to twice the integral from t = 0 to t = x of sqrt(1 + t^4) minus twice the integral from t = 0 to t = x of 1/sqrt(1 + t^4). This all implies that 3 times the integral from t = 0 to t = x of sqrt(1 + t^4), which is the integral you want to find, is equal to x·sqrt(1 + x^4) plus twice the integral from t = 0 to t = x of 1/sqrt(1 + t^4). In evaluating the integral from t = 0 to t = x of sqrt(1 + t^4), one can change variables t = is to obtain equivalence to the integral from s = 0 to s = -ix of i/sqrt(1 - s^4). 1 - s^4 = (1 - s^2)(1 + s^2) = (1 - s^2){1 - (is)^2}. Since the integral from s = 0 to s = u of 1/sqrt[{1 - (ks)^2}(1 - s^2)] is equal to F(u; k), the incomplete elliptic integral of the first kind, this implies the integral just derived is equal to i·F(-ix; i). By changing s = -r, one can also show that this integral is also equal to -i·F(ix; i). Therefore, the integral from t = 0 to t = x of sqrt(1 + t^4) is equal to [x·sqrt(1 + x^4) - 2i·F(ix; i)]/3.
Whats with ∫xⁿ⁻¹e⁻ˣdx and the Gamma Function?
Здравствуйте, пожалуйста что ты называешь "erf(X)" и "erfi(X)" ?
How can the logarithmic intregal (#4) exist for x>1? There is a discontinuity at 1.
I was expecting Chebyshev polynomials, spherical harmonics, or orthogonal functions. Still good stuff tho
Duncan W Those do not really involve integration, though.
@@angelmendez-rivera351 yeah but they're traditionally (at least at my university) part of a class called special functions
Duncan W While they are common special functions indeed, the title of the video was referring to a specific class of special functions, those used in integration.
Can we differentiate Ei(x)+C and get back at e^x/x
are sine and cosine special functions?
I'm confused. Are there infinitely many nonelementary integrals, or is that unknown??
James Oldfield you are correct. There are infinitely many non elementary antiderivatives. In fact, if you just get a random function out of all possible functions, the probability it has an elementary antiderivative is really low, which implies that there are many more non elementary antiderivatives then there are elementary ones. We are just more familiar with the elementary ones.
@@fanyfan7466
How are you supposed to know whether an integral is elementary or not??
James Oldfield it’s actually pretty hard to know for sure. The best thing to do I’ve found is know the famous ones in this video and if you can somehow transform an integral into one of these (through substitution, integration by parts, or something like that) then you know it’s non-elementary. It is quite hard to know if you just grab a random function tho, sometimes you just gotta use wolfram alpha or smth
James Oldfield To know whether a function has a non-elementary integral, use some algorithm, like the Risch algorithm, for instance.
our maths lecturer asked us to evaluate C(x) using Laplace transform and then ended up changing the whole question cos he confused himself🤦♂️
Isnt there a proof that the first integral is not possible unless it is evaluated from -infinity to positive infinity?
If you look at Peyam’s video (English version) and @blackpenredpen video (Chinese version), there’s a proof of how to get sqrt(pi) as an answer for this since it’s a Gaussian integral.
There is no such proof. You can obtain said integral with the error function.
∫xtanx dx is this non-elementary too?
I just noticed your stroke order for boxes is similar to hànzì
Is there a special function that can be used to integrate x^x?
MegaTitan64 x^x = e^[x·ln(x)]. Let w = x·ln(x) = ln(x)·e^ln(x) => W(w) = ln(x) => e^W(w) = x, where W(w) is the principal branch of the Lambert W multi-function. The principal branch is itself a well-defined standard function which is widely used in many fields, from biology to physics and even medicine. Thus, dx = e^W(w)·W'(w) dw = w·(dln[W(w)]/dw)·dw, hence e^[x·ln(x)] dx = (we^w)(dln[W(w)]/dw) dw. As you can imagine, one can integrate by parts to get we^w·ln[W(w)] - Antiderivative{(w + 1)e^w·ln[W(w)]}. w = W(w)·e^W(w) => ln(w) = ln[W(w)] + W(w) => ln[W(w)] = ln(w) - W(w), so the above equals (we^w)[ln(w) - W(w)] - Antiderivative{(w + 1)e^w·(ln(w) - W(w)). The antiderivative of (w + 1)e^w·ln(w) can be computed by integrating by parts, integrating (w + 1)e^w to we^w and differentiating ln(w) to 1/w, giving that this antiderivative is equal to we^w·ln(w) - e^w + C. All that remains to be simplified is the antiderivative of (w + 1)e^w·W(w). To recaputilate, the antiderivative of we^w·W'(w)/W(w), which is what you wanted to find, is equal to e^w - we^w·W(w) + G(w), where G(w) is the antiderivative of (w + 1)e^w·W(w). Unfortunately, there is no actual way to simplify this any further, and there is no special function you can use to obtain G(w).
Not oficially. A Lost proporsal was to name as Sophomore Function.
ILY BPRP
Thank you Oscar! : )
A T Shirt of that Functions Will be great 😄
i ever hear about Lambert (x) and Lambert W(x), is it also from non-elementary integral or what?
You can check out my video on that. X^x=2
How do we evaluate these functions?
Tim Cheung Numerical methods, series, derivable identities, easily calculable tables, etc.
@@angelmendez-rivera351 Do you recommend to use taylor series?
Tim Cheung Yes.
@@angelmendez-rivera351 Thanks
Or maybe Simpson’s rule.
4:54 doesn’t that look like the combination nCr? nCr = (n!)/(r!*(n-r)!)
sir, prove y=mx+c by vector method please
What do you mean "prove"? Do you mean you want to get a line equation from two vectors? We can do that...
if you have two vectors v and w. There is a line that passes over both vectors. We will find useful to define d = (w - v), does not matter which vector you choose to be w or v. Then the slope is m = d.y/d.x - observer that if we flip the vectors in the definition of d, we flips the sign in both numerator and denominator, resulting in the same m.
Wait, what happens when w.x = v.x? Then your neat line equation does not work, because it is a vertical line.
What about c? Well, it is the vertical value when x = 0. We move the vector w, in the direction given by d, until we get x = 0. That is, w.x + d.x*t = 0. We cal solve for t: -w.x/d.x = t, and use it to get c = w.y + d.y*t
Thus, we have
m = (d.y/d.x)
m = (w.y - v.y)/(w.x - v.x)
c = w.y + d.y*t
c = w.y + d.y*(-w.x/d.x)
c = w.y - d.y*(w.x/d.x)
c = w.y - (w.y - v.y)*(w.x/(w.x - v.x))
c = w.y - w.x(w.y - v.y)/(w.x - v.x)
c = w.y - w.xm
Thus:
y = mx + c : m = (w.y - v.y)/(w.x - v.x); c = w.y - w.xm
y = mx + w.y - w.xm : m = (w.y - v.y)/(w.x - v.x)
y = (x - w.x)m + w.y : m = (w.y - v.y)/(w.x - v.x)
y = (x - w.x)(w.y - v.y)/(w.x - v.x) + w.y
If you plug in 1 to Ei(x) shouldn't that diverge?
will newman Yes. The integral is only well-defined for x < 0. The actual definition of Ei(x) utilizes that Cauchy principal value of said integral, defining it for all nonzero real numbers.
@@angelmendez-rivera351 oh I see. Thank you
is integral x^x has a special fonction to solve it
We just say that the integral of x^x is non-elementary.
@@justabunga1 why is it that special ?
@@xaxuser5033 he did this in the video about 2 years ago. You can check out Fematika on his RUclips channel to see how to integrate x^x from 0 to 1.
There is no special function you can use to express the non-elementary integral of x^x.
What about e^cos(x) ?
6:40 the 0 changed
How can i know a certain function Has no elementary anti-derivative?
Risch Algorithm.
Nice hair cut nice integrals
Oh the dread when I first found out about the error function.
I thought you can use u sub on number 2?
There are more function like integration of x^1/2•sinx
bless
Is it possible to do integral x^x?
It’s not since it’s non-elementary.
Not at all, not even with special functions. The best you can do is an infinite series, whose terms in the infinite sum are themselves only expressible as a summation.
Nice haircut....looks aalot like my haircut couple months ago
can you explain about xi function
How to solve: pi^(x^2)=pi•x^2. ?
The answer is plus or minus 1. We cannot solve the equation by hand, but there is a special case for this, which is called the Lambert W function, or product log function if you want to see how they did it in Wolfram Alpha.
π^(x^2) = (e^ln(π))^(χ^2) = e^[ln(π)·x^2] = πχ^2 implies 1 = [πχ^2]e^[-ln(π)χ^2] implies -ln(π)/π = [-ln(π)χ^2]e^[-ln(π)χ^2] implies W[-ln(π)/π] = -ln(π)χ^2 implies x^2 = -W[-ln(π)/π]/ln(π).
Nooooooo!
What are you done?!
That new haircut!!
what about x^x?
Don't forget the +b
I mean +c
What about sinh x/x?
10:32 The main dish.
How old are you?
Мне не нравится псевдослучайной число RND есть , что-то революционное ?
I like that haircut :3
blacksinsredsins
Appeared!!
Fresnel Integrals Are so imigamiary
these are real life horseshoe integrals