Integral of sqrt(sin(x)) vs integral of sqrt(sin^-1(x))

[@alexdemoura9972]

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.

[@alexdemoura9972]

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.

[@alexdemoura9972]

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

[@alexdemoura9972]

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

[@DiegoMathemagician]

@@alexdemoura9972 thanks for the clarification

[@blackpenredpen]

Thanks for the awesome explanations!

[@julius9055]

I share that man's pain of trying to write on a whiteboard left-handed.

[@MuPrimeMath]

Awesome! I would make a post about it but I don't have the community tab yet lol

[@ralfbodemann1542]

2 excellent teachers in one video. You're spoiling us. I'm deeply impressed by the clear and instructive presentation of Mu Prime math.

[@grumpy2606]

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.

[@slowfreq]

Thank you!!

[@andrewdotsonexplains6948]

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

[@raviupadhyay5989]

Can you make a video on why we cannot write Roman number for negative numbers and for zero.

[@arnaudvanuden9495]

Could you please do a video on Elliptic integrals? Very good video btw!!

[@abhishektyagi4428]

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

[@AlBoulley]

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!

[@CrimsonJoker641]

Where can you get the +C shirt, I'd love to buy one!

[@blackpenredpen]

Thanks. It’s in the description

[@iamimran5880]

Both of you are best teacher

[@raviupadhyay5989]

Sir your way of explanation is very nice

[@blackpenredpen]

Could you guess the mysterious guys from the thumbnail?

[@VibingMath]

Of coz he is favourite Mu Prime

[@mohameduvais1821]

Surely

[@alexdemoura9972]

It is a traffic sign, "BPRP Math battle ahead, don't forget +C". The usual fine if the driver forget the "+C" is $750.

[@leif1075]

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

[@gomatichoudhary5537]

You can help me in IIT JEE Exam!!!

[@DSN.001]

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...

[@sunilparekh4581]

A 0.75

[@Pklrs]

can you do the graph of the integral of sqrt(sinx)?

[@mahananda_maiti]

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

[@mahananda_maiti]

@ Yes.

[@sunilparekh4581]

What is E there is it a function, pls tell 🙏🙏

[@mahananda_maiti]

@@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.

[@mahananda_maiti]

@@sunilparekh4581 en.wikipedia.org/wiki/Exponential_integral#:~:text=In%20mathematics%2C%20the%20exponential%20integral,exponential%20function%20and%20its%20argument.

[@AlgyCuber]

mathematicians : makes more special functions also mathematicians : uhh we ran out of names, lets use chinese characters

[@VibingMath]

Glad to see this kind of integral battles with Mu Prime! It's gonna be exciting! Go subscribe Mu Prime!

[@blackpenredpen]

Thank you!!

[@VibingMath]

@@blackpenredpen Welcome!

[@antoniomartelio741]

what is the name of the video mu prime of elliptic integral do not find

[@alessandrovillanuevacantil9618]

Great video, but the guy missed a important parenthesis at the end, i loved it

[@zz4fff]

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.

[@blackpenredpen]

This should help kzbin.info/www/bejne/iX7ViKCHjq9-etE and notice that 2=e^(ln2)

[@rogerramirez266]

muchas gracias , me gusto el video saludos desde Perú :D

[@MrConverse]

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!

[@VibingMath]

Here you are kzbin.info/www/bejne/aHqQkIaMbciqqdk

[@michaelschneider4093]

I bought ashirt and hoodie from you Please make more!!! I got the derivative and integral one.

[@blackpenredpen]

Michael Schneider thank you for your support!! And I will!!!

[@michaelschneider4093]

@@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.

[@michaelschneider4093]

@@blackpenredpen first term calculus student and i love your videos. Dont understand anything but soon!

[@blackpenredpen]

Michael Schneider Thank you!!!! Best of luck on your studying!

[@michaelschneider4093]

@@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!

[@angelmendez-rivera351]

Awesome, but can you do the antiderivative of cbrt(arcsin(x)) and cbrt(sin(x))? :)

[@blackpenredpen]

Lolllll

[@forklift1712]

Why do you have a clock on the whiteboard?

[@GreenMeansGOF]

A couple parenthases are missing at the end but still an awesome video.

[@leif1075]

Sin of u squared equals Sine of 2 over piano..is that what that capital S stands for?

[@justabunga1]

The capital S in S(x) stands for the Fresnel sine integral.

[@leif1075]

@@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

[@justabunga1]

@@leif1075 anything that's a non-elementary function is usually not shown up in test questions.

[@angelmendez-rivera351]

Leif He did explain it in the video. And he made a video talking about this function very recently too.

[@aleksandervadla4840]

Amazing

[@pietroagasi6194]

Hi, where can I find lots of exercises about integrals? (like your 100 integrals video)

[@Patapom3]

Amazing!

[@shiro2738]

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 :)

[@andy-p3d2i]

For x large they are almost the same, so by the Fundamental Theorem of Engineering they are both elementary and have a simple answer :)

[@angelmendez-rivera351]

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.

[@angelmendez-rivera351]

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.

[@sinosodialajay797]

Both are Clever men

[@johnsannicolas2015]

Wow this is so alien compared to how we were taught!

[@HitmanHimself]

Sir can you help me solve integral ((x^n) -1) / (x - 1) limits 0 to 1

[@angelmendez-rivera351]

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.

[@migtrewornan8085]

what's all this "and" crap, the output of "and" is true or false

[@neuralwarp]

Do you mean | ? That's "or" in some programming languages, but in statistics it denotes an event conditional upon another event.

[@migtrewornan8085]

@@neuralwarp Yeah but that new guy keeps saying "and", wtf?

[@karolakkolo123]

@@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

[@angelmendez-rivera351]

Mig Trewornan Uh, no.

[@DiegoMathemagician]

how do you prove a function is non-elementary? how do you even know it for sure?

[@hgnb1001]

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

[@DiegoMathemagician]

@@hgnb1001 Wow it looks hard

[@DiegoMathemagician]

@@cotj332 thanks for sharing, didn't know Risch's algorithm

[@alexdemoura9972]

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.

[@felixangelsanchezmendez1466]

What's the integral of ln(cosx)? Does somebody know? I think it's non elementary

[@angelmendez-rivera351]

Angel Mendes It is indeed non-elementary, and there is no special function for it either.

[@angelmendez-rivera351]

Jayant Singh That gets you nowhere, though, unless you can say what is the integral of xtan(x)

[@ernestschoenmakers8181]

@@angelmendez-rivera351 Also the integral of x*tan(x) is non-elementary.

[@neuralwarp]

Where are S, E, and F defined?

[@LUISFARIASPINO]

X2

[@justabunga1]

Those are something that has to do with an elliptic integral whether it’s complete or incomplete of first and/or second kind.

[@angelmendez-rivera351]

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.

[@torontovlr]

PLEAAASE do 100 Limits in one take, I promise I will watch the whole video

[@vidhipandya2621]

What is the integration of ln(2ix^×^3)/[Sin(ix^×^3)]......

[@angelmendez-rivera351]

Vidhi Pandya What is this expression even supposed to be? Is the inside tetration? What is it?

[@vidhipandya2621]

@@angelmendez-rivera351 Yes,inside it tetration and 'i' is imaginary number and X^x^3 is x raised to the power x cube.

[@angelmendez-rivera351]

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.

[@ReDevil67]

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.

[@mathmathician8250]

Is x|m means x divided by m?

[@alexdemoura9972]

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.

[@jzanimates2352]

Clicked straight away Also, when I clicked on it said 1 view and 4 likes? Maths is even more complicated than I thought . . .

[@solarisone1082]

That's because likes show up faster than views. It's been like that for years.

[@ignore5147]

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 😓🙏

[@sunilparekh4581]

Do you know meaning of E there , is it a function 🙄🙄🙄

[@matteoturrina4317]

@@sunilparekh4581 what do E(t/m) and F(t/m) means?

[@dzundzanify]

OMG I did understand nothing... Where does these values of sin and "u" gets from?

[@angelmendez-rivera351]

Dmitriy Dzundza They are not values, they are substitutions.

[@МихаилСерафимович-в9ф]

Was ist das S(t)?

[@justabunga1]

Михаил Серафимович it’s a sine Fresnel integral.

[@arturovinassalazar]

Help, What is E?

[@justabunga1]

E stands for the incomplete elliptic integral of first kind.

[@jonathangrey6354]

Elliptic integrals???

[@jpradeesh3800]

Why do you mostly solve only integrals?? 🤔

[@blackpenredpen]

Because I love them!

[@mjz5853]

曹老师新发型好帅哦

[@nuklearboysymbiote]

I am loving this collaboration! Any chance anyone else could participate?

[@blackpenredpen]

Sure. Are you interested?

[@nuklearboysymbiote]

@@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

[@blackpenredpen]

@@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 kzbin.info/www/bejne/jqO7YmWvZdWforM then you will know how I used to be : )

[@nuklearboysymbiote]

@@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!

[@BaoNguyen6742]

Can you make n.sinx=x

[@angelmendez-rivera351]

Bảo Nguyễn What? What is this problem? What am I supposed to do? Solve it? And what is n here? Is . standing for multplication?

[@BaoNguyen6742]

Yes and n is a real number

[@angelmendez-rivera351]

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.

[@jamesstewart5062]

Nice vdo sir..... Jai hind 🇮🇳🇮🇳🇮🇳

[@ritikakainth4500]

I don't know why but the 2nd guy kinda looked like harry potter. ( without his glasses 🤣).....

[@dynmo959]

😂

[@sinosodialajay797]

Solve a question using dancing and singing. Pikachu once more.👍👍👍😊😊😊😊😄😄😄😁😁😁😁😎😎😎

[@Agyaatr108]

This Video wanna makes me reply to my own comment

[@Agyaatr108]

Same

[@rajns8643]

Same

[@blackpenredpen]

Same

[@NoNameAtAll2]

Same

[@ayushjuvekar]

Same

[@NoNameAtAll2]

Integral of 1/sin when?

[@madmuffin2511]

With sin-1(x) he means the invers of sin(x) so arcsin(x) = sin-1(x)

[@メ乇しム尺]

@@madmuffin2511 I think he meant: When will bprp do the integral of 1/sinx?

[@karolakkolo123]

@@メ乇しム尺 1/sinx = cscx. Have been done on the channel at least once already, if not more times

[@メ乇しム尺]

@@karolakkolo123 I of course know that, I was only correcting Mad Muffin because I thought he misunderstood what NoName said.

[@justabunga1]

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.

[@ssdd9911]

it would suck if u don't explain the E & F functions and y they have 2 inputs

[@alexdemoura9972]

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.

[@justabunga1]

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.

[@claireli88]

Are both of you friends?

[@ליאורמלכה-ע9ש]

is it just me or he is saing chan lu instade of chain rule?

[@galbi85]

Indeed. He says "Chen lu" on purpose, just joking 😄 There is also the "Prada lu", which is the product rule (for derivation).

[@nathemyaakoubi7511]

Wtf is E

[@sinosodialajay797]

Is he your friend

[@CloudchaserA]

La frustración es muy poca xd

[@emmanuelburetah4579]

It's cool lol

[@chouaiblkdev2720]

I'm the first person to see this video.

[@cristianovipin6842]

Sir ham India'

[@dynmo959]

yo ji mera pand

[@simon56799]

first >:)

[@Peter_Schluss-Mit-Lustig]

To get 10+10 wrong