He did the derivative!!! Check out Fematika • Checking BlackPenRedPe... 100 integrals: • 100 integrals (world r... shop calculus t-shirts and hoodies: teespring.com/... Yes, I am serious. blackpenredpen
thanks , the integral is really amazing. so intelligent. Mathematics is deeply bottomless, and probability and statistics are more difficult to understand. Gamma distribution, chi-square distribution, is so hard to understand.
I was wondering this myself lol at the beginning of this video. I'm betting it inevitably leads to just letting u= the n-th root of tan(x) and then integrating the resulting rational function using partial fraction decomposition - maybe synthetic division for powers of tangent not = 1 or 2.
This should be easy. Separate tan^n into tan^(n-2) and tan^2 then integrate by parts. Then substitute sec^2-1 for tan^2 and you get a reduction formula. Evaluate integral of tan^2 and that should be enough I guess.(I havent tried this but I think this works for even exponents only.)
Evolution can you give me more letters for my substitions into different integral worlds? Sure blackpenredpen ACTUALLY USES OLD LETTERS FOR SUBSTITUTIONS LIKE A BOSS
when you say "please do not ask me to check the answer by diferentiation", I'm pretty sure you were really thinking "please ask me to check the answer by diferentiation" ... So, in order to please you..... Would you check the answer by diferentiation???? Please :)
scrolled down expecting the comment "check the answer by differentiation" as top or only comment .. urs was 3rd and has a bunch of text around it internet, you disappoint...... (altho the expectation ur not meeting, is actually... so in a way.... yay! ....)
This is probably the most fun I've ever had watching a video about integrals! It was a long journey, but it was worth every second. Thankyou for such a great video!
@@blackpenredpen But why on earth would you do that second substitution at all? Isn't there another way maybe more intuitive..like integration by parts Ibwas thinking..Hope you can respond when you can.
16:56 Yeah, putting down some TNT on that would definitely help :) 18:18 It's quite interesting that `1/2` and `√3/2` appear in the completed square, because: a) they are the real and imaginary part of the cube root of `-1` that appear in the complex factorization of the denominator; b) they are the sine and cosine of the 60° angle at which this cube root lays with respect to the real axis :) c) we've got the cube root (of the tangent) in the original integrand. 21:30 We have to go deeper... :> 23:52 The dream is collapsing :J
if they´re not nice they get this integral on their exam! how about that? or worse sin(x)/x from 0 to infinity (but don´t ask for the steps. that´s overkill. though maybe you can show us that one? i didn´t see a single video on YT about that ntegral i would have understood. and the only idea to even make this possible would be using the mclaurin series of the sine)
This channel is now offically a MATH MEME ! "How about the integral of [fancy variation of sqrt(tan x)] ?" - bprp, you have a long meme carrier ahead...
I thought he will forget the + C part Truly professional to the core I am starting to fall in love with math and pens And of course our one and only *CALCULUS*🥳🥳
The second integral can be done without any substitution: Multiply and divide by 2 (you can factor out the divide by 2 later, with the final constant distribution): (2t+2)/(t²-t+1) The derivative of the denominator is 2t-1. So, write the integral as: (2t-1+3)/(t²-t+1) Split the fraction: (2t-1)/(t²-t+1) + 3/(t²-t+1) The first one's integral is ln|t²-t+1|. For the second one, complete the square as (t-1/2)²+(√3/2)². The integral is 2√3*invtan((2t-1)/√3). Now, distribute the divide by 2: 1/2*ln|t²-t+1|+√3*invtan((2t-1)/√3).
I want to see this too, although I'll try it myself first because it may be possible that it be not expressible in terms of elementary functions or with a finite definite expression.
...And when you want to take a break from listening to this video, you are going to SUBSTITUTE IT for one of the easier/more beautiful ones to understand :D Like sin(z)=2
This couldn't have been uploaded at any time better, I'm just about to fall asleep and I always trance at your videos which makes me go to sleep! Thank you so much! :D
So how does the monster grow in the complex domain? I would love to see the secrets in there so that I might apply them to quantum mechanics (and my cool design for a new warp drive model).
i never´ ve seen a intgral like this . i been watching all your videos and in my opinion is very useful. in other side i want to thank, you videos are helping me to improve my english how you know ,y native language is spanish, but thanks for share it. congrats
This is way late, but you can further compactify the answer (and show off some extra algebra skills) by taking advantage of log properties like so: 1/2ln[ sqrt( tan^4/3 x - tan^2/3 x + 1 ) / ( tan^2/3 x + 1 ) ] + sqrt(3) arctan( (2 tan^2/3 x - 1)/sqrt(3) ) + C.
@22:30 - how lazy by resorting to a formula?! Bah! It's another substitution. Let w = (sqrt 3)/2 tan theta. When you substitute for w^2, you can factor out the 3/4, leaving (tan^2 theta + 1) in the denominator, which is a Pythagorean identity. The numerator dw becomes (sqrt 3)/2 sec^2 theta dtheta. Some nice canceling leads to 2/sqrt3 integral of dtheta,... So, 2/sqrt 3 times theta. And, from the substitution I suggested, 2w/sqrt3 = tan theta, so theta = inversetan (2w/sqrt3). So much more elegant not to rely on any formulas beyond the basic integrals.
"and then we have to get back to the U world, and then we have to go back to the X world." What a wild ride - we're going on interplanetary journeys solving this integral!
I kept on making mistakes when trying to check this answer by differentiation, and had pledged to keep this video tab open until I had successfully checked it. So pleased to say that I finally got it right today! I can close the tab now. :D
Small technical error: In the final function, you used the constant C to represent the constant of integration. BUT, you already used C to represent a different constant in your partial fraction decomposition. So, the final solution should use a different constant, like C-sub-1 or K.
