Integral of sqrt(sin(x)) vs integral of sqrt(sin^-1(x))
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- Опубліковано 7 лют 2025
- Integral of sqrt(sin(x)) by using the Elliptic Integral of the First Kind demonstrated by Mu Prime math! Check out his channel: • Crazy limit - This loo...
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I believe this video make many doubts raised about "non-elementary functions", or "special functions" as usually called here in BPRP.
First, function is a relation between sets that associates every element in a set to one element in another set.
Elementary function is a function composed by a FINITE number of operations and/or functions such as exponentials, logs, constants, trigonometric, etc.
By inference, we may assume the "non-elementary functions", or "special functions", are those with INFINITE components of that kind.
Some "non-elementary functions", or "special functions", were shown in previous BPRP videos. Usually they are used to solve Integrals, despite the solution is not undefined, it cannot be expressed by a simple elementary (finite) function, as we are used to. Such Integrals may even have a numerical value as a solution from, for example, an infinite series.
Since Integrals can be a fine expression of a series, a Summation of infinite dx-width elements, a non-elementary function can be another Integral. Mathematics have many non-elementary functions already standardized, such as those used here in this video.
The Fresnel (sine) Integral S(x) is a transcendental function used in optics, closely related to the error function (erf), another non-elementary function.
S(x) = ∫₀ˣ sin(t²) dt
Elliptic integrals originally are used in problems with an arc length of an ellipse. Mathematics defines an "elliptic integral" as any function which can be expressed in a form containing a rational function of two arguments: a polynomial of degree 3 or 4, and a constant. In general, they cannot be expressed in terms of elementary functions. In this video there are two Elliptic integrals.
Incomplete elliptic integral of the first kind:
F(t, k) = F(t | m=k²) = F(sin t, k) = ∫ dt/√(1 - m.sin²(t))
Incomplete elliptic integral of the second kind:
E(t, k) = E(t | m=k²) = E(sin t, k) = ∫ √(1 - m.sin²(t)) dt
@@alexdemoura9972 thanks for the clarification
Thanks for the awesome explanations!
I share that man's pain of trying to write on a whiteboard left-handed.
Awesome! I would make a post about it but I don't have the community tab yet lol
2 excellent teachers in one video. You're spoiling us. I'm deeply impressed by the clear and instructive presentation of Mu Prime math.
Very informative and insightful. Entertaining video. Thank you for covering such relatively uncommon and interesting integrals in general. It is always entertaining to watch your videos.
Thank you!!
Can you make a video on why we cannot write Roman number for negative numbers and for zero.
I was messing around on WolframAlpha and found this interesting integral sin(x)^(3/2). I would really appreciate it if you could do a video on that integral
mathematicians : makes more special functions
also mathematicians : uhh we ran out of names, lets use chinese characters
Sir your way of explanation is very nice
Both of you are best teacher
Could you guess the mysterious guys from the thumbnail?
Of coz he is favourite Mu Prime
Surely
It is a traffic sign, "BPRP Math battle ahead, don't forget +C". The usual fine if the driver forget the "+C" is $750.
What the he'll is that E?? Does,that stand,for e the base ofbthe natural log?? Or what..so you cant solve this without knowing that special function..??. Not a question of intelligence
You can help me in IIT JEE Exam!!!
I don't know why but the 2nd guy kinda looked like harry potter. ( without his glasses 🤣).....
😂
Apologies if this has already been asked, but is pi / 4 + x / 2 preferred over (pi + 2x) / 4, when writing in "simplified" notation? The latter takes less space to write, and arguably "looks" simpler too. Thanks for your guest solve this BPRP video!
Could you please do a video on Elliptic integrals? Very good video btw!!
Love you sir. I am student of 18 years , but I watch your every video. Your video help me to improve my skill in Calculus and also help to I.A. Maron.
Really
@ Yes.
What is E there is it a function, pls tell 🙏🙏
@@sunilparekh4581 In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument.
@@sunilparekh4581 en.wikipedia.org/wiki/Exponential_integral#:~:text=In%20mathematics%2C%20the%20exponential%20integral,exponential%20function%20and%20its%20argument.
I don't know if you already tried this or you just don't care about the idea, but I was thinking that you could make a video comparing these two integrals:
∫x/(sqrt(x^4+10x^2-96x-71))dx
∫x/(sqrt(x^4+10x^2-96x-72))dx
The thing is, even when they look almost the same, one of them is elementary, and the other is not :)
For x large they are almost the same, so by the Fundamental Theorem of Engineering they are both elementary and have a simple answer :)
Andrei Secuiu They are looking for exact answers here, not engineering memes. Also, they are both certainly non-elementary. This can be proven relatively trivially.
x^4 + ax^2 + bx + c = x^4 + ax^2 + a^2/4 + bx + c - a^2/4 = (x^2 + a/2)^2 + bx + (c - a^2/4) = (x^2 + a/2)^2 + 2ux^2 + au + u^2 - 2ux^2 - au - u^2 + bx + (c - a^2/4) = [(x^2 + a/2)^2 + 2u(x^2 + a/2) + u^2] - [2ux^2 - bx + (au + u^2 + a^2/2 - c)] = [x^2 + (a/2 + u)]^2 - [sqrt(2u)·x - b/{2·sqrt(2u)}]^2 = [x^2 + sqrt(2u)·x + (a/2 + u - b/{2·sqrt(2u)})][x^2 - sqrt(2u)·x + (a/2 + u + b/{2·sqrt(2u)})], where u is a solution of the equation 8u^3 + 8au^2 + (2a^2 - 8c)u - b^2 = 0. The square root of the expression just derived is equal to the denominator of the integrand. If sqrt(2u) = 0, then the expression is identical to completing the square on the denominator, which then yields an integrand with an elementary antiderivative, which can be antidifferentiated by letting y = x^2 + a/2. Otherwise, this antiderivative is non-elementary. However, sqrt(2u) = 0 implies b = 0. Since your integrals do not satisfy this, they both clearly are non-elementary. You can also simply choose to verify using Wolfram Alpha, but I am simply giving the actual mathematical proof here.
what is the midmost number between 0.5 and 1?
a) arithmetic mean, 0.75
b) geometric mean, 0.707106...
c) harmonic mean, 0.6666...
d) agm, 0.728395...
e) quadratic mean, 0.790569...
A 0.75
Is x|m means x divided by m?
You are not the only one with such doubts. I posted a comment (with 3 replies), it contains the definitions used in this video. It is in Comments section. Please check if it could be of any help.
Where can you get the +C shirt, I'd love to buy one!
Thanks. It’s in the description
I bought ashirt and hoodie from you
Please make more!!!
I got the derivative and integral one.
Michael Schneider thank you for your support!! And I will!!!
@@blackpenredpen please do! I remember a few months ago you had some cool ones on the store. I didnt see them this time. But im anticipating my order.
@@blackpenredpen first term calculus student and i love your videos. Dont understand anything but soon!
Michael Schneider
Thank you!!!! Best of luck on your studying!
@@blackpenredpen thank you! I dont know if you know physics or linear algebra but if you can explain some of those concepts that would be awesome!
SIR THE RESOURCES AND LINKS TO LEARN MATHEMATICS THAT YOU SAID IN YOUR VIDEO WITH fematika ARE STILL NOT UPLOADED IN THE DESCRIPTION OF THE VIDEO , please do upload those links
Great video, but the guy missed a important parenthesis at the end, i loved it
Could you please do a video on the whole “DI method” of integration? Honestly, I’m pretty sure you’ve already done one, so perhaps a suggestion: could you please put a title card in the upper right of your videos linking to the one on the DI method whenever you use it in another video. It would help!
Here you are ua-cam.com/video/2I-_SV8cwsw/v-deo.html
Glad to see this kind of integral battles with Mu Prime! It's gonna be exciting! Go subscribe Mu Prime!
Thank you!!
@@blackpenredpen Welcome!
A couple parenthases are missing at the end but still an awesome video.
muchas gracias , me gusto el video saludos desde Perú :D
what is the name of the video mu prime of elliptic integral do not
find
What's the integral of ln(cosx)? Does somebody know? I think it's non elementary
Angel Mendes It is indeed non-elementary, and there is no special function for it either.
Jayant Singh That gets you nowhere, though, unless you can say what is the integral of xtan(x)
@@angelmendez-rivera351 Also the integral of x*tan(x) is non-elementary.
Now integrate x*sec(x) dx
can you do the graph of the integral of sqrt(sinx)?
Terrific! Amazing video, but when I try to solve an integral like 2^x ln x Wolfram Alpha give me Ei(x) function as a solution. That's ok! What I want to know is how from original integral I can get that special function. Please, video links or references for intermediate steps.
This should help ua-cam.com/video/SMrWmQYJKDo/v-deo.html
and notice that 2=e^(ln2)
Sin of u squared equals Sine of 2 over piano..is that what that capital S stands for?
The capital S in S(x) stands for the Fresnel sine integral.
@@justabunga1 thanks why doesn't he explained that in the video..so there's no way to solvw it without just knowing that fact then...not a good test question theb
@@leif1075 anything that's a non-elementary function is usually not shown up in test questions.
Leif He did explain it in the video. And he made a video talking about this function very recently too.
Wow this is so alien compared to how we were taught!
Awesome, but can you do the antiderivative of cbrt(arcsin(x)) and cbrt(sin(x))? :)
Lolllll
Amazing!
Both are Clever men
Where are S, E, and F defined?
X2
Those are something that has to do with an elliptic integral whether it’s complete or incomplete of first and/or second kind.
S(x) is defined as the sqrt(2/π) times the integral of sin(πx^2/2) with x running from 0 to x, while E is the incomplete elliptic integral of the second kind, and F is the incomplete elliptic integral of the second kind. Difficult to explain what they are here, so you should probably read the Wikipedia article on those two.
PLEAAASE do 100 Limits in one take, I promise I will watch the whole video
Amazing
Why do you have a clock on the whiteboard?
Hi, where can I find lots of exercises about integrals? (like your 100 integrals video)
what's all this "and" crap, the output of "and" is true or false
Do you mean | ?
That's "or" in some programming languages, but in statistics it denotes an event conditional upon another event.
@@neuralwarp Yeah but that new guy keeps saying "and", wtf?
@@migtrewornan8085 | in elliptic integrals or hypergeometric functions is just a separator, which doesn't differ in functionality from a simple comma. The notation is different for reasons unknown to me, but that's how it is usually notated for those types of special functions
Mig Trewornan Uh, no.
Wtf is E
how do you prove a function is non-elementary? how do you even know it for sure?
It's hard to prove, basically you have to transform your function into one that it's already proved to be a non elementary one. You can read about in this paper: math.dartmouth.edu/~dana/bookspapers/elementary.pdf
@@hgnb1001 Wow it looks hard
@@cotj332 thanks for sharing, didn't know Risch's algorithm
You are not the only one with such doubts. I posted a comment (with 3 replies), it contains the definitions used in this video. It is in Comments section. Please check if it could be of any help.
Hello Blackpenredpen, I am finishing my degree in my college and am working on a personal project that can redefine our understanding of time and space. Unfortunately, however, I am limited by my inability to solve some calculations as I study physics, but the math involved is extremely complex, unlike anything you have ever seen. My work consists of 4 fundamental equations, which become more than 20 equations, of the most diverse types, differential equations, integrals of various variables, summation of differential equations, among others. After solving these 4 equations, the work just begins, but it would be helpful to have a brilliant mind like yours working on it. If you are interested I can send you one of the equations for you to try to solve.
Clicked straight away
Also, when I clicked on it said 1 view and 4 likes? Maths is even more complicated than I thought . . .
That's because likes show up faster than views. It's been like that for years.
Sir can you help me solve integral ((x^n) -1) / (x - 1) limits 0 to 1
HitmanHimself (x^n - 1)/(x - 1) = 1 + x + ••• + x^(n - 1). Integrate this expression from 0 to 1 to get C + x + x^2/2 + ••• + x^n/n evaluated from 0 to 1. The expression at x = 0 is C, while at x = 1, it equals C + 1 + 1/2 + ••• + 1/n, which implies the difference is equal to 1 + 1/2 + ••• + 1/n, which is equal to H(n), the nth Harmonic number.
Please guide me
Why integration of (√1-sin^2.t)=E(t/m)+C and
Integration of 1/(√1-sin^2.t)=F(t/m)+C
😓🙏
Do you know meaning of E there , is it a function 🙄🙄🙄
@@sunilparekh4581 what do E(t/m) and F(t/m) means?
Was ist das S(t)?
Михаил Серафимович it’s a sine Fresnel integral.
OMG I did understand nothing... Where does these values of sin and "u" gets from?
Dmitriy Dzundza They are not values, they are substitutions.
曹老师新发型好帅哦
Elliptic integrals???
Help, What is E?
E stands for the incomplete elliptic integral of first kind.
This Video wanna makes me reply to my own comment
Same
Same
Same
Same
Same
I am loving this collaboration! Any chance anyone else could participate?
Sure. Are you interested?
@@blackpenredpen I don't think I am advanced enough as I am still in sixth form (equivalent to high school), but I am currently learning differential equations and I have tried my hand at MIT integration bee questions. I also like working with systems of linear equations with matrices. I have uploaded videos but they are unlisted bc i am shy haha. But if there's anything that I can learn to do in short time and you want to collaborate I think the shyness can be overcome!
I also speak Cantonese and Mandarin just fyi
@@nuklearboysymbiote Ooh ok!! You can start by practice some on your own. Just start recording and practice talking to the camera. Take a look at my first ever video on YT ua-cam.com/video/XrX12y0pllQ/v-deo.html then you will know how I used to be : )
@@blackpenredpen oh I see haha! Actually I just had a maths test on friday and last night I recorded just under 1 hour of me going through the test and explaining the steps. I'll edit the video and send you a link (might contain one or two curse words because I made the video with the intent of sharing between my friends, hope thats ok with u). Then let's see if we could work together!
What is the integration of ln(2ix^×^3)/[Sin(ix^×^3)]......
Vidhi Pandya What is this expression even supposed to be? Is the inside tetration? What is it?
@@angelmendez-rivera351
Yes,inside it tetration and 'i' is imaginary number and X^x^3 is x raised to the power x cube.
Vidhi Pandya Okay then. ln(2ix^(x^3))/sin(ix^(x^3)) = [ln(2i) + x^3·ln(x)]/[i·sinh(x^(x^3))] = [ln(2i)/i]/sinh[x^(x^3)] - i·x^3·ln(x)/sinh[x^(x^3)]. Once you reach this step, it becomes clear that there is no way to integrate this... not even using special functions.
Why do you mostly solve only integrals?? 🤔
Because I love them!
Solve a question using dancing and singing.
Pikachu once more.👍👍👍😊😊😊😊😄😄😄😁😁😁😁😎😎😎
Can you make n.sinx=x
Bảo Nguyễn What? What is this problem? What am I supposed to do? Solve it? And what is n here? Is . standing for multplication?
Yes and n is a real number
Use r instead of n to signify real number, since n signifies natural number, or in some cases, an integer. With that said, r·sin(x) = x is not actually a solvable equation. This because the inverse function of x |-> r·sin(x) - x is not expressible with elementary functions. In fact, for certain values of r, the inverse does not even exist.
it would suck if u don't explain the E & F functions and y they have 2 inputs
You are not the only one with such doubts. I posted a comment (with 3 replies), it contains the definitions used in this video. It is in Comments section. Please check if it could be of any help.
The E and F are both elliptic integrals. E stands for the incomplete elliptic integral of second kind, and F is for incomplete first kind.
Nice vdo sir..... Jai hind 🇮🇳🇮🇳🇮🇳
Are both of you friends?
Integral of 1/sin when?
With sin-1(x) he means the invers of sin(x) so arcsin(x) = sin-1(x)
@@madmuffin2511 I think he meant: When will bprp do the integral of 1/sinx?
@@メ乇しム尺 1/sinx = cscx. Have been done on the channel at least once already, if not more times
@@karolakkolo123 I of course know that, I was only correcting Mad Muffin because I thought he misunderstood what NoName said.
NoName the notation for this is another way to write as arcsin(x). This is not the same as 1/sin(x). The notation and the exponent looks weird the way it was in calculators or textbooks. 1/sin(x) is the same as csc(x). He already did the video of integral of csc(x), which is -ln(abs(csc(x)+cot(x)))+C. Another answer can be ln(abs(csc(x)-cot(x)))+C.
is it just me or he is saing chan lu instade of chain rule?
Indeed. He says "Chen lu" on purpose, just joking 😄 There is also the "Prada lu", which is the product rule (for derivation).
Is he your friend
La frustración es muy poca xd
I'm the first person to see this video.
Sir ham India'
It's cool lol
yo ji mera pand
first >:)
To get 10+10 wrong