Elementary vs. Non-Elementary integral battles! (beyond regular calculus)
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- Опубликовано: 2 сен 2019
- Integration techniques required: integration by parts, u sub, trig sub, partial fraction, algebra, experience, and patience.
Check out these 8 special functions: • how WolframAlpha defin...
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Battle 1, integral of cos(x^2) vs integral of cos(ln(x)), 1:00
Battle 2, integral of ln(1-x^2) vs integral of ln(1-e^x), 7:55
Battle 3, integral of x^(x/ln(x)) vs integral of x^x, 16:23
Battle 4, integral of x*sqrt(x^3+4) vs integral of x*sqrt(x^4+4), 19:29
Battle 5, integral of x/ln(x) vs integral of ln(x)/x, 32:25
Battle 6, integral of ln(ln(x)) vs integral of sqrt(x*sqrt(x)), 34:00
Battle 7, integral of sqrt(sin(x)) vs integral of sin(sqrt(x)), 36:13
Battle 8, integral of sqrt(tan(x)) vs integral of tan(sqrt(x)), 40:52
Battle 9, integral of tan^-1(x) vs integral of sin^-1(x)/cos^-1(x), 59:13
Battle 10, integral of 1/(1-x^2)^(2/3) vs integral of 1/(1-x^2)^(3/2), 1:04:23
subscribe to @blackpenredpen for more integration videos.
Battle 1, integral of cos(x^2) vs integral of cos(ln(x)), @1:00
Battle 2, integral of ln(1-x^2) vs integral of ln(1-e^x), @7:55
Battle 3, integral of x^(x/ln(x)) vs integral of x^x, @16:23
Battle 4, integral of x*sqrt(x^3+4) vs integral of x*sqrt(x^4+4), @19:29
Battle 5, integral of x/ln(x) vs integral of ln(x)/x, @32:25
Battle 6, integral of ln(ln(x)) vs integral of sqrt(x*sqrt(x)), @34:00
Battle 7, integral of sqrt(sin(x)) vs integral of sin(sqrt(x)), @36:13
Battle 8, integral of sqrt(tan(x)) vs integral of tan(sqrt(x)), @40:52
Battle 9, integral of tan^-1(x) vs integral of sin^-1(x)/cos^-1(x), @59:13
Battle 10, integral of 1/(1-x^2)^(2/3) vs integral of 1/(1-x^2)^(3/2), @1:04:23
file: docs.wixstatic.com/ugd/287ba5_3f60c34605f1494498f02a83c2e62b29.pdf
New challange for me😊
wow nice timestamp! Should be pinned yrself!
Where are those special functions?
Yale NG Which ones are you talking about? They never appeared in the video.
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
Another approach to the integral of ln(1-x^2) dx would be to factor the inside and then use the product rule of logarithms to get the integral of ln(1-x) + ln(1+x) dx. It's a bit easier to solve this way.
Solution to integral of sqrt(tan(x)):
There's a blackpenredpen video on that + c
Just a real minor point of #4: you could also do a hyperbolic trig substitution instead, and you'd get a simple inverse hyperbolic sine term in the final answer instead of the natural logarithm. That natural logarithm is also convertible to the inverse hyperbolic sine.
The way I like to think about the Integral of cos(x^2): with some clever substitutions and Euler's formula it can be shown that it can be written in terms of the integral of e^(x^2) and since that cannot be defined in terms of elementary functions, thus the integral of cos(x^2) cannot be
heloo
I love your enthusiasm!
I love these videos! Encore, encore :-)
So nice to see a notification from bprp just after the first day of school :D
Thanks!!!!
I want to know, how to prove that the integral of a function is not elementary, please tell
Chirayu Jain
It’s quite hard to prove it mathematically. I think we need to know Galois theory from advanced abstract algebra in order to do so. I actually don’t have experience in it unfortunately.
@@blackpenredpen, what a coincidence I started learning abstract algebra just 2 weeks before., 😁
Galios Theory, it's probably easier to just know which ones are non-elementary, rather than to prove each one individually.
Chirayu Jain You can prove the non-elementariness of an integral using the Risch algorithm.
@@japotillor That by itself doesn't disprove that there might be some weird unknown way to do an integral.
One-hour long video but u definitely spent a lot more time than that! Your effort should be appreciated! And also the patreon list grows longer everytime 😁👍
PS it's 1am here in HK and yr thumbnail looks cool with some chill 😆
Mak Vinci lollll thank you!! I prob will make another thumbnail tho. I don’t think that is that appealing lol
@@blackpenredpen Hey keep this kind of thumbnail man(but not too many), it makes others curious to press the thumbnail 😁
You are a great teacher
who else got a smile on the face at 16:15 because you have watched an old bprp video?
Sun and clouds me
Sun and clouds Nah, I still messed it up, ffs. 😂😂😂
Take my love for this channel from Bangladesh.
To integrate arcsin(x)/arccos(x) from x = -1 to x = t < 1, let x = cos(θ). Then dx = -sin(θ) dθ. The integrand is now -arcsin(cos(θ))·sin(θ)/θ. The bounds are from θ = π to θ = arccos(t). On the interval (0, π), which is the codomain and range of arccos(t), arcsin(cos(θ)) = π/2 - θ. Therefore, the integrand is -(π/2 - θ)·sin(θ)/θ. Factoring -1 will change the bounds to run from θ = arccos(t) to θ = π, with integrand (π/2 - θ)·sin(θ)/θ. By linearity, this gives the integrals of (π/2)·sin(θ)/θ and -sin(θ). The first integral is equal to (π/2)·(Si(π) - Si(arccos(t))), and the second is equal to cos(π) - cos(arccos(t)) = -(1 + t). Then the total integral is simply equal to [(π/2)·Si(π) - 1] - (t + Si[arccos(t)]). Call (π/2)·Si(π) - 1 = C, so the integral is simply C - t - Si(arccos(t)). Done! For the record, Si(x) is defined as the integral from s = 0 to s = x of sin(s)/s.
We can extend the answer to other intervals, but this requires some caution, since arcsin(cos(θ)) = π/2 - θ is no longer true in other intervals.
22:21 Euler's substitution sqrt(u^2+4)=t-u would be better idea here
Last one third Euler substution (with roots) or integrating by parts also are good option
This is the best video You have made - of those I've seen.
I was especially happy to know that ln(ln(x)) is a non-fundamental function. That question has been bothering me for years.
On the first one, it was obvious, because cos(ln x)=(x^i+x^-i)/2. Power rule, separate real and imaginary coefficients, and put it back to trig functions. Even if you're not going to use complex numbers, you can guess the right integral because cos is like an exponential and goes well with ln and poorly with x^2.
Brilliant sir
The second one was a bit over the top, ln(1-x^2)=ln((1-x)(1+x))=ln(1-x)+ln(1+x)
Either way you need to do integration by parts. Personally, I broke up the ln but if makes sense to use IBP with a bit of work extra then go for it. As long as you get an answer and understand the process
GhostyOcean no you don’t need to do integration by parts with the method he stated. After you split the ln you can split the integral and solve them both by u sub
-James- Integrating ln(u) requires integration by parts, so you are wrong.
@@-james-8343 in order to integrate ln(x) you need to do IBP unless you have the answer memorized (xln(x)-x)
Gábor Tóth Tbh, it’s just as hard if you split it. I split it, and if anything, that made it harder because you have to do IBP twice.
Integration of e^-xx from +inf
To -inf with pler co-ordinates
15:15 „Integrale für Euch“ 😂
Grüße an alle Deutsche 🇩🇪🙌🏽
SGE 1899 Hahahah yea!!! Lars helped me to translate it. : )
hahahah Ehrenmann
one take, with some cuts. i dig it 😁
I came here after the rap battle in 8 Miles😂... I am ready for the battle!!!
Lets see those special functions!
19:57 you can do both u-sub and trig-sub at the same time by letting x^2=2tan(theta) ;) then, xdx is nicely equal to sec^2 and the rest is just the usual
LETS GO!
Best videos sir for maths
15:05 In the last two terms of that answer (before the +C) it was not necessary to use absolute value around the ln input. Respond to this comment if you can figure out why!
Hey BPRP , can you make a video about Group Theory ?
What font did you use in your document? Do you use LaTeX package or?
Big salutation from Algeria thank you Allah blesses you
√tanx i love this integral same as 1/(x^6+1)
7:55 wouldn't that be easier to just factor 1-x^2 as (1-x)(1+x) and then use the log propertry to split the ln of the product?
Number 9 is a pretty straightforward battle, once you know the formula for antiderivatives of inverse functions. As long as a function has an elementary antiderivative, its inverse has an antiderivative of the form, xf^-1(x) - F(f^-1(x)). Once you know tan(x) has antiderivative ln|sec(x)| + C, you just plug tan^-1(x) into the formula and do some trig identities on sec(tan^-1(x)) to get the same result.
Thumbnails are getting stronger
which integrals are intermediate and high school?
Would it actually be faster to integrate cos(ln(x)) by using the complex definition of the cosine? You would then need to integrate (x^i+x^-i)/2, which is just a matter of integrating polinomials.
Account Fantoccio Relatively, yes.
integrating ln(cos x) would be an interesting one
Please make a collaboration video with 3blue1brown together
YES
YES
@@azujy2959 gosh that would be so cool
Number 8 was crazy
Hey Im a Calculus amateur. Just wondering what method did bprp used at 38:50. Thx in advance!
can show hyperbolic functions more love or not?
Did You make already any video with non-elementary integrals like eliptic ones?
You probably made that future video already, but it is interesting to point out that the most obvious attempt to antidifferentiate arcsin(x)/arccos(x) with respect to x results in the sine integral:
A basic trigonometric identity has arcsin(x)=π/2−arccos(x), from which the integrand becomes ½π/arccos(x)−1; then the substitution x=cos(y) with dx=−sin(y)dy results in the sine integral.
That is, ∫arcsin(x)/arccos(x) dx = -x−½π∫sin(y)/y dy = −x−½πSi(arccos(x))+C.
On question number (8). Suppose you let integral equal to Q, then square both sides and integrate twice then take the sqr,, can it work?
It will be a great pleasure to me, if you explain how to separate elementary from nonelementary ones. Does such formular exist?
Промо Risch algorithm.
Sir please make a video on ramanujan formula on finding value of pi
Oon Han has made a video on it
In India we have National Teachers' Day on 5th Sept. So, Happy Teachers' Day to BPRP and all other teachers in advance.
Now solve the special function ones!
Hey bprp, what font do you use in your files and thumbnails?
integrating arcsinx/arccosx is actually doable;much easier to do than the other ones mentioned as undoable previously. its just a bit of subs and ibp and using the Si function.
We presently scratch the integral, if it is a non-elementary integral.
can you know if the integration is possible or not just by looking at it ? , and if yes how do you know ?
May the chenlu be with your integrals.
Battle 2: Don't use partial fraction! Use ln(ab) = lna + lnb rule first, much more simple!
That was my thought. ln(1+x) + ln(1-x)
That's what I did - got two standard ones.
Correct me if I am wrong, but at 8:50, you can factor 1-x^2 and use rule of log to expand it into 2 terms?
Oh yes. Then integration by parts after that. Both work
@@blackpenredpen Right, unless one memorize that integral of ln(x) is xln(x)-x hehe
n choose k yea
How to do that (long division)?
Hi BPRP, and thank you for the videos :D I guess this comment will go unnoticed, but if I never ask, I'll never know :)
Why are half of these functions impossible to integrate? You just mention as a fact that it's impossible but never why. I'm not great at integration, so I don't understand _why_
Sir ,What is the integral of ∫(1-x^2)^n dx
Isn't it instead of using partial fractions, Can we not have
xln (1-x²) -2x + tanh-¹ (x) +c ?
As the answer?
I like your microphone
Sir, why don't you make a video about proving that the ramanujan formula
the ad I had for this just said "Find your Steve" 😱😱😱
!!!
Here's another way to write the answer to question 2, xln(1-x^2)-2x+2tanh^-1(x)+C
Wait... 1 hour 😯💚
Question 3, the absolute troll
Second round:
integral 1 / (1-x^2) = arctanh x + C
I never forget the chendu😆
salut monsieur svp j'aimerais avoir un pdf des 100 integrale ou un pdf d'çntegrale pour licence de mathematiues svp
Yall I was just vibing to the Doraemon theme song in the beginning.
12:30 you could just directly integrate it to 2tanh^-1(x).
instead of partial fractions.
حودا روك No, because the domain would be incorrect.
What about x^dx? Can u do ir pls?
kemosabe What is that?
m8 im in high school learning quadratics XD
could u do a video where u explain calculus and why it works sorry i just kinda don't get what ur doing and just don't get calculus - but i still sub
Its all about analyzing a graph of the function. Integral is giving u a surface area under a function. Derivative is the gradient of a line tangent to the function
@@LeeSeungrhee yes i got the practical part but the theory is really confusing (actual formulas etc)
Can you teach us group theory?
hello brother. I get a different answer for number 2 intergral ln(1-x^2)dx instead of 1-x i get x-1 and 1+x is same as x+1
On 14 September it is teacher's day in India . Please make a excellent special video on the day.
How we can know what is elementary and what is not?
Only between you and me!😁
Isn’t it easier on the 2nd one to change it from ln(1-x^2) to ln((1-x)(1+x))=ln(1-x) + ln(1+x) and integrate like that?
Jack Hounsom Eh... it's about as easy, but it depends
Jack Hounsom Nah, it’s worse, I did it, and trust me, it’s worse.
58:05 is just insane lol
n choose k yea! And I didn’t do partial fractions just to save time. Lol
For no. 8, can't we split 1/(t^2-2) into partial fractions and use ln? It is much friendlier than coth. Also, why coth instead of tanh?
Yes. But it would be just longer...
@@blackpenredpen But why coth instead of say tanh? According to you they are identical...
Mohammad Zuhair Khan ln in this situation is not friendlier than ln, since the inside of ln would be a complicated expression. In fact, coth is expressible in terms of ln, so that makes your point moot.
blackpenredpen Tbf, I prefer it because you can see how you get the answer, whereas the tanh is just a standard formula.
How do you make your thumbnail🙏😊
I use “page” on Mac, math type and pictures.
blackpenredpen can you please give any suggestions for android phone or windows laptop as we don’t have an MacBooks or IPhones or iPads with us.
Good video, can you please help me with this integral
.. X*Sec(X)
Integration by parts
X take D and I sec x
Integration of secx is log|secx + tan x| and then its easy
It's non-elementary because if you try to do IBP, you get xln(abs(sec(x)+tan(x)))-integral of ln(abs(sec(x)+tan(x)))dx. Here integral of ln(abs(sec(x)+tan(x))) is non-elementary.
Can you solve it
Int. (x-2)/[(x-2)^2(x+3)^7]^1/3
26:25 100 Integrals #61.
bprp: *showing 8 integral battle*
me: ...here we go again
Konstantin Cherkai 10*
Could you solve this integral? Integral of (secx)^(3/2). I wish you did it. Thanks for giving a lot of support
Angel Mendes This integral is non-elementary, so there is no solution anyone can give you that you would be satisfied with.
Thank you so much, bro
For #9, the ln part turned out to be ln|cos(arctan(x))|, anyone else have this??
Freddie Correct
Wow
Why IS integral of tan (sqrt x ) impossible to solve
I genuinely don't understand
Battle 8 is the best integral....
Almost an f-bomb at 27:35!
Do you think that Isaac newton would have been able to derive all of these integral solutions back in his day
for number 2, isn’t the int of 2/1-x^2 just 2arccot(x)?
Yes.
It’s 2artanh(x), like the hyperbolic inverse tanh function
How did he found out that we can't do the other one?
11:22-11:25 the integral of the thing you are saying needs partial fractions doesn't, actually, because the answer is clearly inverse hyperbolic tangent (Argthx/Argtanhx)
Serouj Ghazarian Well, that's not correct either, since the domain or arctangent is different from the domain of the function we started with. Strictly speaking, partial fractions are the only correct way to get the most general antiderivative, and this can be proven.
@@angelmendez-rivera351 ArGtanH, not arctan
@@angelmendez-rivera351 the function we started with is ln(1-x^2), which has EXACTLY the same domain as Argtanh.
Lol, I speak Irish but I don't know if that helps in the slightest
Hi, cos(X square) is a function . Geogebra gives a result, if you integrate ( calculate the area) between 2 points
Why we can say that this integral does not have a result.thank you For your reply
Cent Uğurdağ Because the antiderivative of cos(x^2) is *not* the area. The antiderivative of cos(x^2) is simply another function, but the area under the curve is a number. Not remotely the same thing. Any software can calculate any area, but if you ask Geogebra to give you the antiderivative, it *cannot* and *will not* give you an answer, because there is no answer.
İ agree but want to know why there is no antiderivative of this function
BPRP is an asmr youtuber now? 58:30
Yeah mate, he’s done it before.
Number 2 way easier to write ln(1-x^2)=ln(1+x)+ln(1-x)
BPRP how to find range of Sinx-√3.cosx+1
There's a formula for turning a linear combination of sin and cos into a single sin (or cos) with a phase shift and coefficient. Then you just need to adjust the range for adding 1.