integral of sqrt(x^2+1), with Euler Substitution, math for fun

[@tokenup420]

trigsubs are way easier typically but I may do it this way in my D.E. class just to make my teacher think im smarter lol.

[@blackpenredpen]

that works!

[@holyshit922]

Here after Euler's substitution and some linear properties you will get integal of power function To see that not always trig subsittutions are faster calculate integral Int(\frac{dx}{x^2(4x^2-3)^2\sqrt{x^2-1}}) or Int(\frac{dx}{x\sqrt{2x^2-2x+1}}) If we use first Euler substitution for integral Int(\frac{dx}{x^2(4x^2-3)^2\sqrt{x^2-1}}) we will get integral which can be easily calculated mentally but if we use third substitution (with the roots) we should use Ostrogradsky method for isolating rational part of integral and calculate twelve coefficients If you still want trig substitutions you need three substitutions (actually two of them are inverse trig substitution)

[@holyshit922]

Yeah is there someting easier then power rule and linearity of an integral , because it we will get after Euler substitution in this integral

[@Linus-ex3xs]

Jacek Soplica you do LaTeX right?

[@BY-sh6gt]

I use multiple u-subs on the exam and got the exact same answer... but that's too much subs and after i submitted my answer i just realised that trig-sub 'exists' 😂

[@swedishmathtutor6104]

And all of you hating and saying why don't you do trig sub. This is the beauty of math. 1. Two different methods arrive at the same solution and 2. There exists a much more elegant approach. Now imagine this was the way you were taught and someone showed you trig sub. What would you say?

[@marcushendriksen8415]

"That's pretty awesome!" (but then again, I do appreciate the fact that there are multiple ways to skin mathematical cats)

[@arjavgarg5801]

Swedish Math Tutor I would say, why didn’t you tell me this first?

[@peorakef]

id say: you cheater!

[@SimsHacks]

Also, this sub works always, not the case with trig sub, where you have troubles with domain if you put x=tan(theta) for example.

[@g0rgth3b0rg]

Rewriting (1/t^2 - t^2) as a difference of two squares, then multiplying top and bottom of the fraction by the conjugate, only to create a difference of two square conjugates is beautiful and amazingly clever.

[@blackpenredpen]

Greg Brown thank you!!!!

[@allaincumming6313]

Sí, se soltó la greña lmL

[@ekxo1126]

@@allaincumming6313 "Yes, the lmL hair was released" thanks google translate

[@farazriyaz9078]

@@ekxo1126 I wonder where can I buy the rumored lmL hair...

[@otromenfeick9382]

@@ekxo1126 I'm a native spanish speaker, that's indeed what it means. The commentary itself is what doesn't make sense lol

[@mohan153doshi]

This Euler's sub method of solving this rather simple integral was really fascinating and I just loved the algebra involved here. The ugly radicals just vanish like pure magic. Who said that Math is not magic? You just proved that math can be magical and beautiful. Thanks for this awesome substitution. It can make anyone's day (anyone who likes math) beautiful. In a parallel universe, I hope I come across a math teacher like you in real life instead of the virtual interaction here. I would surely feel blessed then.

[@jif7707]

What's is the "ugly radical"?

[@maalikserebryakov]

@@jif7707 ur mum

[@toopytoopy8547]

*Try to subtitute x by the hyperblic sinus : sinh(t)* as a remind, sinh(t) = (exp(t)-exp(-t))/2 ; cosh(t) = (exp(t)+exp(-t))/2 ; sinh'(t)= cosh(t) then if x = sinh(t) , we have t=ln(x+sqrt(1+x²)) _(easy to obtain in two lines!)_ So, *1+x² = 1+sinh(t)² = cosh(t)²* and *dx= cosh(t) dt* let's note int(f(x)dx ,0 ,a ) the integral of a function f(x) between 0 and a then int(sqrt(1+x²) dx ,0,a) = int( cosh(t)² dt, 0, ln(a+sqrt(1+a²)) *## i will note b := ln(a+sqrt(1+a²) which means that sinh(b) = a##* But we have cosh(t)² = (exp(2t)+exp(-2t)+2) / 4 so int( cosh(t)² dt, 0, b) = [exp(2t)/8 - exp(-2t) /8 + t/2] # between 0 and b = exp(2b)/8 - exp(-2b)/8 +b/2 Moreover exp(2b)-exp(-2b) = (exp(b)+exp(-b)) (exp(b)-exp(-b)) = 4 sinh(b) cosh(b) and we know that cosh(b) = sqrt(1+sinh(b)²) then int( cosh(t)² dt, 0, b) = ( sinh(b) . sqrt(1+sinh(b)²) ) /2 + b/2 . As we have a = sinh(b) then : int( cosh(t)² dt, 0, b) = (a . sqrt(1+a²))/2 + ln(a+sqrt(1+a²))/2 Which means that the integral function of sqrt(1+x²) is *(x sqrt(1+x²))/2 + ln(x+sqrt(1+x²))/2* AS FOUND IN THE VIDEO Notice that if you deal well with hyberbolic trigonometry this way of calculating the integral is faster and easier !!

[@allaincumming6313]

In this case: Hyperbolic sub>Trig sub>Euler sub

[@samevanovitch3362]

You are amazing and I was wondering why he didn't use the change X = SINH(t)

[@최문규-o4d]

Wow U are genius

[@Rosie6857]

Another substitution that works is x = tan t

[@peamutbubber]

Nah tany = x is way faster

[@neomooooo]

Excellent video! I prefer x=sinh(t). Hyperbolic trig sub is an elegant approach, and is a middle ground between Eulers trick, and classic trig sub.

[@maalikserebryakov]

Not really

[@DougCube]

Two minor issues... 1) You never justify swapping the order of the absolute-value and -1 power at 17:12. 2) At 19:38, the reason why the inside is positive and you can drop the absolute-value is not complete. If you have (A+B) and A > B, you cannot conclude (A+B) > 0. Counterexample: A=1, B=-2. (The missing reason is that if x is negative, the A portion is greater than -x to get the sum to be positive.)

[@antimatter2376]

1st the reason he can brirng the -1 to the t as a power is because of a ln property. 2nd it's different because you square x and add 1 inside a square root and subtract x. if you just have sqrt(x^2)-x then it's zero. But instead we have the +1 so sqrt(x^2+1)-x is always positive. You're example doesn't consider the square and sqrt.

[@Jamelele]

when do you specifically need to use euler's sub? That'd be interesting :)

[@blackpenredpen]

It was a question sent by one of my subscribers

[@maalikserebryakov]

You can use it whenever there is a specific composition of a quadratic inside a radical But when the quadratic contains the x term, trigsub will no longer work. So you will use euler sub then.

[@sandile13809]

This made me appreciate trig sub

[@johngreen3543]

Trig subs can be avoided in many cases. Particularly with radical expressions with x2 - 1, x^2 +1 and 1-x^2 in the radicand. Do not use trig for them

[@marvalmej007]

Genuinely nice explanation! This is amazing. Math is beautiful and magic, you are an amazing magician. Kudos!!

[@diamonddave2622]

looks like we are now black pen, red pen and blue pen!

[@blackpenredpen]

Diamond Dave yup!

[@papajack2205]

Diamond Dave there was already a video with a green pen included. I guess crazy things are going on these days..

[@AlgyCuber]

purple pen too

[@slippygames3519]

@@AlgyCuber blackpenredpenbluepengreenpenpurplepen

[@ellyvatedaf]

@@slippygames3519 RGB pen

[@NonTwinBrothers]

Every time I see Euler in the title I'm like "Daaaamn he EVEN has a substitution????"

[@lebgdu17pktudorpa23]

Little note : 1-t^2 / 2t , is actually the formula (when you let t=tan(theta) ) for 1/tan(2*theta) . This method thus shares a relation with trig sub anyway :) Very nice video!

[@complex314i]

t^-2-t^2 = (t^-1-t)(t^-1+t) = 2x(t^-1+t) = 2x(t^-1-t+2t) = 2x(2x+2t) = 4x(x+t) = 4x(x^21)^(1/2) Easier to do difference of squares while still in t.

[@gloystar]

Ohh! Thank you so much for this video that answered my question. Good job bro, keep it up!

[@isobar5857]

Geez...you have the patience of a saint!

[@goedelite]

The frequency of advertising interruptions for an educational feature is disgraceful. I would remember who the advertisers are - not to buy their products but to avoid them!

[@anoopkumar-dt7wp]

We had a sir who taught it like this x/2(question)+constant/2(integral of reciprocal of question)

[@johnmahoney5805]

Never thought I'd say that I'd prefer to do trigonometric substitution, but here we are.

[@maalikserebryakov]

Trig sub is not useful when there are many other terms apart from the radical This method works always to simplify the integrand

[@schoob69]

Just an awestruck highschool calc student here, but, at 14:32, why do the conjugates cancel in the denominator? I get that x^2 + 1 - x^2 equals one but what about the second sq root (x^2 + 1) + x term? What happens to that?

[@johngreen3543]

The best method is not either trig sub or euler sub. The best method is integration by parts. Let u =(x^2+1)^1/2 and dv = dx. then a little add and subtract will give the original integral plus a familiar integral. Give it a try.

[@zoso25]

While making the initial substitution for X+t, couldn't we just integrate Xdx and tdx in parts and then substitute dx with the dt form? That would simplify things I guess. We won't need to substitute t with X form for atleast one part. I'm probably not thinking this through and might be overlooking something.

[@mbulut7]

I was going to say to the publisher, "hey bro, you just missed the beauty of this method: first part is just (x^2)/2, and you do just t*dx part". I said let me first scan through the comments not to repeat the same thing. then I saw your comment. You are damn right, that is the way to do this integration.

[@holyshit922]

No you should express x and sqrt as a function of t and then differentiate x with respect of t

[@SimsHacks]

We learn it since it is universal. Trig sub doesn't always work if you have other things apart the sqrt in the integral.

[@maalikserebryakov]

I see. Thank you. This will help me immensely

[@bmrm2004]

easier way: just replace x with sinh (t)

[@nejlaakyuz4025]

Or tan(t)

[@1_adityasingh]

@@nejlaakyuz4025 then you have to integrate sec (x)

[@me_hanics]

@@1_adityasingh integrate of secant is "known", this channel has a video on it

[@tikz.-3738]

@@me_hanics it's ln(tan(X)+sec(X)) it's a pretty standard integral should be remembered probably

[@digroot]

If x^2 + 2 then your method might not work if you sub sinh(x), Euler method seems to be working in x^2 + a, maybe x^3 + a, not sure yet

[@lemonlimeGOD]

Wow, you were right. That WAS really cool!

[@blackpenredpen]

Alex Behlen I am glad that you like it!!!

[@janderson2709]

Why does the initial equation equal X+t?

[@JoshuaHillerup]

Jacob Anderson he just declared it is. You're allowed to do that when you are defining a new variable t.

[@erinasama4976]

Jacob Anderson because it must be something bigger than x

[@sherllymentalism4756]

Even negatively bigger 😂

[@neuralwarp]

The wording would be "select t such that sqrt(x2+1) = x+t".

[@holyshit922]

In polish and russian schools Euler's substitutions was standard ones f.e can be found in russian textbook Курс дифференциального и интегрального исчисления Григо́рий Миха́йлович Фихтенго́льц with short geometric interpretation

[@pedrojosealbarracinvargas7533]

¿cuál de las 3 sustituciones Euler es mejor? ¿algún consejo?

[@MrOLOY-mj8fi]

HI was it possible to simply use x=sh(t) ?

[@IzayoiMeiHigu]

can you make a video demonstrating the Euler substitution??? I don't want to memorize without understanding why is that possible, but I can't find the demonstration on Wikipedia ... :(

[@holyshit922]

Говоришь по русский ? Hablas espanol ? They have recorded video about Euler's substitutions

[@IzayoiMeiHigu]

si hablo español, como lo notaste? :O

[@holyshit922]

Sprichst du deutsch ? I found Euler's algebra book in which is rationalization of square root of quadratic The guy whose video i had watched changed state of his videos to private but russian is even better because you have video on youtube and also quite good book Курс дифференциального и интегрального исчисления Фихтенгольц Г М Euler's book Leonheardi Euleri opera omnia www.math.uni-bielefeld.de/~sieben/Euler_Algebra.ocr.pdf page 349

[@IzayoiMeiHigu]

thanks :D

[@holyshit922]

Watch also separation of rational part of the integral of rational function (Ostrogradsky method of undetermined coeffincients ) because it can be useful after Euler substitution Your analysis of Euler substitution start with cutting curve y^2=ax^2+bx+c with secant line

[@zahari20]

The best way to solve this integral is by setting x = sinh(t). Then we have Integral (cosh(t)^2 dt and here cosh(t)^2 = 1/2(1 + cosh(2t)) etc.

[@chazzaca]

At 17:00 I would have just put in t=sqrt(x^2+1) - x and be done with it!! Other than that great video and thanks for showing us that Euler rules!! :-)

[@ahmeddamour3661]

Chapeau ! 😊 On peut aussi faire X = Tan (téta) ... par la substitution trigonométrique.

[@swedishmathtutor6104]

Never seen that method! Very interesting. Would like to see the idea behind it

[@dalisabe62]

The idea behind it is the assumption that there exists a linear equivalent for any nonlinear one with the incorporation of another variable in the addition format (if you multiply t by x, you only make it more complicated) The question is: under what circumstances this Euler method is more efficient that simple trig substitution? Euler is well-known for creating such equivalence. Consider the famous Euler formula in the complex plane which maps a complex exponential to a complex trigonometric form. That formula was built on the same pattern of logic. Because integration is such an art, transformations are central to finding anti-derivative of complicated integrands.

[@vitalchance5768]

Should I show a real elegant way of taking this integral? I think the way above is really insane. Just for the audience: it should be done in three simple steps just with one substitution. The way Boris Demidovich was doing it in 1970's.

[@johngreen3543]

Demidovich must have come from Eastern European schools as a student in his younger days. Western European instructors seem to rely to heavily on trig sub.

[@OverCookedRice]

When you trying to find the easiest way to integrate a problem and you found this guy who makes it harder. I know how to solve this problem and after I watched this video I dont know how to start anymore.

[@maalikserebryakov]

😂😂😂😂😂😂😂 LMAO

[@maalikserebryakov]

sounds like you have trouble organising your knowledge on integral calculus. This is something I have given a lot of thought to. Want some help?

[@OverCookedRice]

@@maalikserebryakov Hi, thanks for your offer. I’m just joking. I’ve passed the class and moved on for a while. Something I will never look back :)

[@shadowstryder0]

excellent job! this was a beautiful integration.

[@joseantoniogonzalezgarcia3518]

Muy bien, ese cambio de Euler. Otra forma más de para completar la derivada ¡ Le felicito !

[@Billy_98]

You are awesome ,dude.Thx for this video!!

[@nadiralishah6889]

How good you are changing the markers in the hand.

[@mireksoja9063]

I like the video. Good job. Sometimes you could be too fast but it's fine because I can stop video for one or two seconds. Reminds me my old good times when I was a student. :)

[@blackpenredpen]

Mirek Soja thank you. Yea pause the video whenever you need to. Hopefully overall is good.

[@holyshit922]

I think he chose wrong example If he had chosen Int(sqrt(x^2-1),x) as an example he would have shown two Euler substitutions which cover all integrals in the form \int R(x,\sqrt{ax^2+bx+c})dx Substitution with leading coefficient he showed but substitution with the roots is missing If the purpose of this video is calculating this integral then video is ok If the purpose of this video is to show another substitution which is less known in US then video is not finished

[@RubenHogenhout]

This is much more interesting because if you have the equation of the circel Y^2 + X^2 = r^2 then as you write it as a function f(x) = (r^2 -x^2)^(1/2) you can intregrate the Circel if you calculate the intergral.

[@davidseed2939]

At 4:05 you could have made things simpler by separating the integral into intgl( x dx) + intgl( t dx)

[@carce8450]

what an amazing channel i just found !

[@blackpenredpen]

Glad you like it!

[@chungys3660]

I can understand this. No wonder I'm alone.

[@blackpenredpen]

lol

[@wr9411]

It’s not hard to understand, it’s just a substitution with lots of algebra

[@aryanks2167]

we are with you

[@JoshuaHillerup]

Why are you able to assume that X only has real values like that?

[@marcushendriksen8415]

Because he's chosen the reals as the domain for this function. He could have extended it to all complex numbers if he'd wanted

[@danny1504-g2d]

Very cool thank you BPRP!

[@aakashjana6225]

Why do you hold such a big mic cant you use a more compact clip mic

[@hachemimokrane2810]

Can we put x=tan(t) ?

[@johngreen3543]

Should be secant as the radicand is x^2-1.So the answer is NO.

[@jdinitials]

At 12:39, how is it possible to include the square on the numerator from 1/t^2?

[@stefan4321]

Because 1 = 1² 😉

[@peterjivotovski1828]

Would a "u substitution" work too? It seems like a much simpler solution...

[@mickeygrump]

How many of these types of problems would be on a test?

[@theadel8591]

This Euler fella his name‘s all over mathematics

[@MarkPaul1316]

very interesting this substitution for the calculation of this integral. had solved this integral by trigonometric substitution x = tgO.

[@yassine321]

i never knew this method for integration anyway i gain another weapon in my magical pocket Thanks bprp !

[@khiariyoussef6674]

nice !you can also : sub x with tan(u) then you integrate by parts : 1/cos^3(x) !

[@blackpenredpen]

Yup! That's trig sub!

[@fgdhlololo1887]

khiari youssef no parts needed...

[@nickharland6473]

@@fgdhlololo1887 parts is usually used to compute the integral of 1/cos^3(x)

[@daisyypoon]

Then seems to be way easier.

[@ases4320]

Trig subs are way smoothier, are these formulas for paricular cases?

[@JSSTyger]

I'm pretty sure the trig substitution (sec²(x) = 1+tan²(x)) is for this particular case but the Euler substation might give you a general answer for sqrt(x+a).

[@holyshit922]

For this integral Euler subsittution gives integral with less calculations

[@ases4320]

You sure? Maybe i'm too comfort with trig subs, i should practice a little with that to compare

[@joluju2375]

I can't hear the words at 0:41 : "and I'm just going to say that ??? is one".

[@cipherunity]

Fantastic. Is there some other method to do this question?

[@integration_IIT_ISM]

Put x=tan theta.... Then u get integral sec³ theta... Untegrate it by parts

[@voodoo_child01]

You can also solve this really easily by integration by parts, and it also has a shorter answer... please tell me if you want to know the solution.(it involves recursive integrals)

[@osbornelawani6277]

Yes

[@Dharmarajan-ct5ld]

Shouldn't√(x+t)^2 be |x+t| ?

[@asm.ez94]

Hi I have integration I want help from one. How can I communicate?

[@AshishSingh-753]

Hey buddy your are a best mathematics teacher

[@doublestarsystem]

In my opinion, it is more important for students to learn mathematical modelisation of real life problems, including variables identification, and finding out the general equation of a a specific problem, then let the computer do what it was built for: Tedious calculations....(I was able to calculate this integral on my smartphone within 10 seconds using a TI-89 emulator)....I still wonder if there is a physical phenomena that obeys this law ?

[@TheTorito09]

Brillant! Just one thing, did you said in the minute 5:04 "polinomio"? Cause that is not a polinomio by definition... the exponent must be natural number

[@NTPCuber]

can you teach me how to integral this ' integral of( 2/x + √(4/x - 1))^(-1/2) '

[@Bicho04830]

I tried to derive that and I could not get to the integrand. Also, I tried to use the x=tan(t) and x=sinh(t) substitutions to solve the integral and I got stuck. Please help.

[@sherllymentalism4756]

How were you stuck? Maybe we could email

[@ThAlEdison]

hmm if you set x=cot t, then ta n (t/2) = w, you get that x=(1-w^2)/2w So it's equivalent to doing a trig-sub followed by a Weirstrass substitution. Kinda sorta

[@bowenchai8474]

Maybe Integration by parts is another way that can solve in a more faster way,because in the process ,add 1 and minus 1 could make a difference.

[@eliteteamkiller319]

I forgot all about the existence of Euler substitution. Maybe we spent a day on it in class.

[@aryansonthalia9298]

That's really nice. Would you give some questions where I can try to use it

[@TheJaguar1983]

Damn, that's crazy. No wonder I had so much trouble solving this on my own.

[@rounakagarwal5136]

You made this super easy sums look complicated and tough

[@DiegoMathemagician]

PLEASE FACTOR that 1/2 thank you :)

[@sharkknight618]

1/3*3/2

[@digantabaro6572]

Why does the 'x+t' substitution works ?

[@knochiosiedlerfreund255]

well done, Steve. You pronounced Euler in the correct way. :)

[@nathandaniel5451]

Out of curiosity, in what situation would this method be more effective than other methods?

[@holyshit922]

You can see that after this substitution its enough to reverse power rule so here it is effective

[@nathandaniel5451]

@@holyshit922 been awhile since I saw this video, I'm guessing the reason for my comment was that I didn't see a difference in the effectiveness of this technique over trig sub and was wondering about situations where trig sub would be impractical but this is.

[@letslearnmathematics5432]

Why we cant use power rule?

[@tywarwick]

Smart approach but perhaps a little quicker by an 'x = sht' substitution

[@Kalimaco]

Could you show and prove ds^2=1/y^2(dx^2+dy^2)....

[@nor557utakata9]

天才やん

[@mohammadelsayed5715]

Keep going, you’re awesome 🌹

[@digbycrankshaft7572]

Just one thing. At the end when you removed the absolute value signs from sqrt (x^2+1)+x what if you took the negative root of sqrt (x^2+1) then sqrt (x^2+1)+x

[@OnePieceFan4765]

The square root only gives the positive root. You only need to worry about +- when it’s the solution to an equation

[@digbycrankshaft7572]

@@OnePieceFan4765 👍

[@markproulx1472]

What is the object you’re holding?

[@kennkong61]

I'm too lazy to search all the comments on this channel to see if this question has been asked before: Are the presenters Ood?

[@LucasGabriel-lc9ty]

Muito obrigado, acho q agr vou conseguir resolver uma integral q eu tô penando a quase 2,5 semanas com isso que acabo de conhecer (substituição de Euler)

[@juliojossias8200]

Qual?

[@ezeldintayel6247]

I did the integral by using )complex substitution where I put x =isin(theta) and it worked!!

[@giampiproietti]

And that's why the hyperpolic functions exist. You'll get the solution a lot easier with 'em.

[@blackpenredpen]

Gigi 96 yes

[@LucaScarpellini3d]

Does "sqrt" stand for "squirting",right?

[@raduvasilache6804]

easier method: Write sqrt(x^2+1)= (x^2+1)/sqrt(x^2+1). You get a integral that has a simple formula(that ln|x+sqrt(x^2+1)|) and also a integral that you solve using partitions. In final you will get that the double of that integral equals sth you know and it will be that thing/2

[@Magic73805]

Sir, How to solve.. Square root 1+x^4 ???? Please tell me...

[@jdratlif]

This was great. Thanks.

[@ujueije5762]

Can you integrate this integral for me ? (sin(sqrt(x)+a) * e^sqrt(x)) / sqrt(x)

[@rushildalal2974]

Substitute sqrt(x)=t. So, dx/2sqrt(x)= dt. So dx/sqrt(x)=2dt. The integral now is 2[sin(t+a)*e^t]dt. Using integration by parts, let value of integral be I. So, I= 2[sin(t+a)*e^t] - 2[integral(e^t*cos(t+a)dt)]. Using by parts again, I= 2[sin(t+a)*e^t] - 2[cos(t+a)*e^t +{integral(e^t*sin(t+a))}] . However this second integral is what we started with, so its value must be I. So the equation becomes I= 2e^t[sin(t+a)-cos(t+a)] + 2I. Solving, we get I = 2e^t[cos(t+a) - sin(t+a)], t=sqrt(x) So the answer is 2e^sqrt(x)[cos(sqrt(x))-sin(sqrt(x))] + c

[@mohamadsharafeddinbabouji8728]

I do have one question What made you think in the first place that you have to choose x+t instead of only t ?? Otherwise it was amazing algebra and magic way of thinking ☺

[@SimsHacks]

Cause it works. He didn't think of it,it's a well-known sub.

[@박주은-f4x]

Thank you so much!! I love this vedio

[@Treegrower]

Very interesting, I've never seen this method before! However, I think I'll stick to trig-sub. This looks hard!

[@ronvaiz123]

would you show how to integrate sqrt of actanx?

[@NotYourAverageNothing]

You can't. It has no Elementary solution.

[@ronvaiz123]

Ty

[@blackpenredpen]

ron vaizman if u mean sqrt(tan(x))... I might do that one day when I have a much bigger board :)

[@ronvaiz123]

No my friend..i meant to sqrt(tan^-1(x))

[@blackpenredpen]

that is not possible, just like N0tY0ur4v3r4g3N0th1ng said. but sqrt(tan(x)) actually is

[@abdouabdou2647]

Great Mr 👍

[@Kudravets-Diana]

How to do the integral of squre ((0.5)^2+x^2) ?

[@sourinchatterjee597]

What to use by eular method when the integration is (1-x^2)^1/2?

[@holyshit922]

That is what i mean that this video is not finished

[@thatwhichislearnt751]

You can do (1-x^2)^1/2 = (1-x)*t This rationalizes the integrand. Or you can factor out i=(-1)^1/2 and use the same as in the video, the complex numbers disappear at the end, since (1-x^2)^1/2=i*(x^2-1)^1/2

[@holyshit922]

I dont know why you omit integration of irrational functions - so called integration of binomial differentials (Tschebyshov substitutions, Tchebyshov proved that only three cases have elementary antiderivative) - integration functions with square root of quadratic trinomial using Euler substitutions Euler substitutions allow you to find u subsitutions which look like Weierstrass substitution As I mention earlier there are three Euler substitution and this substitution do not cover all cases To cover all cases we need also Euler substitution with the roots sqrt(a(x-x_1)(x-x_2))=(x-x_1)t Substitutions which look like Euler substitution we can get from simple exercise with right triangle Draw right triangle and label it as in invers trig substitution, then draw bisector of angle complementary to theta You will get another triangle after drawing bisector In this new triangle find angle pi/4+theta/2 and calculate its tangent