Hey fellas! =D Spring is offering a 20% discount off EVERYTHING at my Merch shop. Just go over to papaflammy.myteespring.co/ and use code HEATWAVE20 :)
@kentlouisoctaviano4466
Ай бұрын
BRO LOOK AT THE DATE ON YOUR OLD VIDEO "OH NO DADDY" BRO THE DATE IS OLDER THAN KZitem REALEASE💀💀
@leif1075
Ай бұрын
Flammy why not juet rewrite x as (x^2)^1/2 and solve it thst way..Hopenyou can PLEASE sjare why yiu did it this convoluted way. But thanks for sharing.
@alexander8311
Ай бұрын
nice try, but we know that dx is very small, so higher orders are basically zero which yields the trivial result of zero for the integral
@PapaFlammy69
Ай бұрын
fucc, u got me D:
@KinuTheDragon
Ай бұрын
But this isn't (dx)^2, this is d(x^2).
@alexander8311
Ай бұрын
@@KinuTheDragon for dx = x = 0 it holds
@diobrando7642
Ай бұрын
@alexander8311 wait, so dx = sin(x)
@othila9902
Ай бұрын
@@diobrando7642 wrong. dx=dsin(x)
@YellowBunny
Ай бұрын
I'm not familiar with this kind of mathematics, but here's my approach. Let y=x^2, making the integral y^(1/2) dy. This equals 2/3*y^(3/2). Substituting back yields 2/3*y^3. Plug in the bound to get 2/3*(1-0) = 2/3. So, I got the same result. But I'm not sure how well this idea generalizes or if there are any ways in which it breaks.
@shaurryabaheti
Ай бұрын
thats what I did xD
@Qaptyl
Ай бұрын
dx^2 is dx^2/dx * dx = d/dx[x^2] * dx = 2xdx
@ianmathwiz7
Ай бұрын
They are equivalent as long as g is continuously differentiable and strictly increasing.
@mrpsychodeliasmith
Ай бұрын
That's exactly what I did. I put u = x^2, so dx^2 becomes du, and x = u^(1/2). Then integral is u^(1/2) du which gives U^(3/2) / 3/2 => 1/(3/2) i.e. 2/3.
@dank.
Ай бұрын
@@Qaptyl Leibniz notation is the best lol
@phonon1
Ай бұрын
Very useful integral in probability.
@williamnathanael412
Ай бұрын
Help explain
@phonon1
Ай бұрын
@williamnathanael412 In statistics, every random variable admits a cumulative distribution function F_X, but not every random variable admits a density/mass function. So, in general, we can define the expected value of a random variable X as E(X)=\int x dF_X(x) (integrating w.r.t. the cdf).
@phonon1
Ай бұрын
@@williamnathanael412 see properties section in this wiki article as a starting point: en.wikipedia.org/wiki/Expected_value?wprov=sfla1
@diobrando7642
Ай бұрын
@@phonon1isn't the generalized case the integral over Omega of X in dP?
@ekxo1126
Ай бұрын
yes he should also show for a discrete probability how the integral works out to be the usual sum
@mitch523
Ай бұрын
No one gonna talk about the farmers tan?? Goes crazyyyyy
@eigenchannel-137
Ай бұрын
Papa flammy has been engulfed in flamen!
@Daniel-ef6gg
Ай бұрын
From the product rule, d(x*x^2) = x d(x^2) + dx*x^2. So x^3|(0->1) = int(x d(x^2))|(0->1) + int(x^2 dx)|(0->1). 1 = int(x d(x^2))|(0->1) + ⅓. So the answer is 1 - ⅓ = ⅔
@PapaFlammy69
Ай бұрын
Very nice approach! :)
@siwygameplay
Ай бұрын
This is just integration by parts btw
@renanwilliamprado5380
Ай бұрын
@@siwygameplay Not exactly, it depends on your interpretation.
@tfg601
22 күн бұрын
@@renanwilliamprado5380 and what interpretation is that
@jppagetoo
Ай бұрын
Once you defined the terms of the original integral it all made sense. I never thought about integrating across a different function instead of just dx.
@PapaFlammy69
Ай бұрын
:)
@ChrisRossaroDidatticaDigitale
Ай бұрын
12:00, 2i-1.
@ofridaniel2127
Ай бұрын
Yeah i thought i was crazy for a second
@adgalad25
Ай бұрын
It yields the same result. But yeah.. it's 2i-1. I had to do the sum like 5 times to convince me 😂
@ricardoparada5375
Ай бұрын
Oh wow, I wasn’t expecting to see a Riemann-Stieltjes integral today
@nablahnjr.6728
Ай бұрын
nice display of Riemann/Stieltjes methods people just do the last one in practice though, now i wonder if there would be a generalization if g was not continuous
@tomkerruish2982
Ай бұрын
There is; in fact, the definition of the Riemann-Stieltjes integral does not require g(x) to be continuous. It must, however, be of bounded variation, which means it can't be too 'wiggly'.For example, sin(1/x) would be a bad choice in the vicinity of x=0, even if you explicitly exclude x=0. A full explanation is outside the scope of what I can write in a KZitem comment.
@KinuTheDragon
Ай бұрын
@@tomkerruish2982"I have a marvelous proof of this fact that this comment section is too narrow to contain." - Fermat if he used KZitem
@vincentproplayer
Ай бұрын
Isn't this solvable by writing x as sqrt(x^2) and then treating it as int{sqrt(t)dt} with t= x^2
@hassanniaz7583
Ай бұрын
yea that's basically what i thought (to let x^2=u). But this video wasn't just about getting the right answer. This vid provided a good insight on what it means to integrate w.r.t g(x)...
@ZipplyZane
Ай бұрын
@@hassanniaz7583Sure, but I thought he'd use substitution at the end to show how to relate the two different types of integrals.
@Brandon-be2uw
Ай бұрын
Is really similar to the easier way he uses later in the video, but notice that x ≠ sqrt(x²), in fact, sqrt(x²) = |x|. Still, in this case, it is true, since we are working in [0,1], but the correct way to conclude what you said I think it would be Int x dx² Int x 2x dx Using t = x² and dt = 2x dx Int sqrt(t) dt
@digbycrankshaft7572
Ай бұрын
Yes. Exactly how I approached it.
@neboskryobchannel5303
Ай бұрын
Why using change of variables, isn't it just Int 2x^2 dx = 2/3 * x^3 ?
@flamurtarinegjakyt3745
Ай бұрын
Never seen a definite integral done by definition. Very exciting video
@PapaFlammy69
Ай бұрын
thx :)
@Legion22Cl217
Ай бұрын
Personaly, I'd just use some differential geometry result dx² is a volume form onto the segment [0,1], it can simply be calculated as : dx² = 2xdx (as it is the exterior derivative of the scalar field x->x²) Then your integral is simply 2 \int_0^1 x²dx=2/3, that's all folks
@dank.
Ай бұрын
Exterior derivative go brrrrr
@tomfan5863
Ай бұрын
that was my immediate thought. dx^2 is 2 x dx, and then do the integral.
@GeodesicBruh
Ай бұрын
Yuuuup this is the way to do it quickly.
@vincentstone7272
Ай бұрын
I like this way
@gabrieleinsiedel1849
Ай бұрын
19:08 = integrate [cos(x) d(sin(x))] from 0 to 1. 1. Apply previous rule, yielding: integral {cos(x)[sin(x)]' dx} from 0 to 1 2. Since derivative for sin(x) = cos(x), that yields: integral [cos²(x) dx] from 0 to 1 3. Just solve 4. = (1/2) + [sin(2)/4] 😁 Edit: plus sign rectified 😅
@bobdavid01
Ай бұрын
Slight correction, the sin(2)/4 has a positive sign
@discontinuity7526
Ай бұрын
this was exactly the kind of video I needed right now. I've been out of university for a few years and getting back into calculus, and your explanation at the beginning could not have been better for me. I had no idea you could integrate by different functions like that, and I was amazing that the result of xdx^2 was 2/3 haha. Absolute banger and I will definitely be revisiting this multiple times
@georgasatryan3876
26 күн бұрын
just say u= x^2 and you'll have integral(sqrt(u))du what equals ⅔*u^3/2 equals ⅔*(u^½)^3 and since u = x^2, it means sqrt(u) = x so: ⅔*x^3.
@nigerianprinceajani
Ай бұрын
It is dx² = 2xdx, so xdx² = x(2xdx), thus integral(0,1)(xdx²) = integral(0,1)(2x²dx) = 2integral(0,1)(x²dx) = 2(⅓1³ - ⅓0³) = ⅔ Note that in identifying x(2xdx) with 2x²dx I'm using that we have a module operation from the ring of smooth functions on all differential-k-forms defined by left-multiplication.
@gavinh8146
Ай бұрын
At 11:53 he says “2i - i” but means, I think, “2i - 1”. I found this very confusing at first because the “i”s look almost the same as the “1”s on the blackboard.
@fritskuijk
26 күн бұрын
You are right, but whatever comes out of it will be of size n and thus will be ruled out by the n^-3 factor. That might be the reason he did not correct the error if he became aware of it
@bingchilling4717
Ай бұрын
couldnt you switch the variable? if you consider x^2=t that means x=sqrt t since x is possitive so the integral just becomres integral from 0 to1 of sqrt t dt which is 2/3
@kappascopezz5122
Ай бұрын
Commenting from just the thumbnail to say that it makes perfect sense if you just say y=x², implying dy = 2x dx, so dx² = 2x dx. In total: int_0^1 x dx² = int_0^1 x 2x dx = int_0^1 2x² dx = [2/3 x³]_0^1 = ⅔
@threepointone415
Ай бұрын
The Rie-womann Integral
@douglasstrother6584
Ай бұрын
That was fun! Very "Michael Penn".
@ben_adel3437
Ай бұрын
I love evil looking integrals and the fact that we can actually do stuff to calculate them
@taterpun6211
Ай бұрын
Although it might not be obvious at first the point of this integral modification, one of its strengths shines in A(x) being a partial sum function (A(x)=A(floor(x))=sum of terms
@encounteringjack5699
Ай бұрын
Nice! Little piece of new info. I tried it before watching and got the same answer. What I did was I set x^2 as y. Solved for x to get sqrt(y). Didn’t change the bounds cuz it’s just 0 to 1 and those don’t think would really change given this scenario. So now it’s the integral from 0 to 1 of sqrt(y) with respect to y. This gives the answer of 2/3 as well. Playing for a bit, changing the bounds to that function is accurate. If it were 0 to 2, it’d be the integral from 0 to 4 since (0)^2 is 0 and (2)^2 is 4. Comparing that to the form of solving this for continuous functions. Integral from a to b of f times g’ dx. Gets the same answer for the 0 to 2 situation. Which is 16/3.
@kostasch5686
Ай бұрын
The 6 dislikes are probably coming from the very misleading thumbnail featuring Feynman and x*dsinx.
@marekrawluk
Ай бұрын
Yet another way: d(x^2) may be multiplied by 1, where 1 = dx/dx. Next d(x^2) * (1/dx *dx) makes d(x^2)/dx *dx, and this yields to 2x*dx, guessed immediately, waiting impatiently till 18:00. No formal prove, just "an engineer" version, but when our mathematical language uses some basic "grammar" rules it should be created in that way - our convention way. The 2x multiplier makes non-linear expansion on x axis of a regular Riemann integral, showed quickly somewhere in the mid part of the movie.
@Mrlonely345
22 күн бұрын
Its becomes really easy if we write dx^2 = 2xdx . And the answer is 2/3 simple
@tushi10
Ай бұрын
It's simple, you can Wright the x to ((x^)1/2)^2. Then you simply integration it and you will get (2x^3)/3. Then you input the limit and the ans is 2/3.
@-.-Infinity-.-
17 күн бұрын
For context At 2:40 and just a senior at highschool Isn't this the same as Integral of √xdx from 0 to 1 Can't we just extrapolate the answer from that Or let x²=u then it just becomes a normal integral With the last formula, Int cosx d(sinx) = Int cosx × cosx dx = Int cos²x dx = Int (cos2x+1)/2 dx = (1/2) Int (cos2x +1) dx = (1/2) (sin2x/2 + x + C1) = Sin2x/4 + x/2 + C Sorry if I did anything incorrect But is that really the only way to do that, i feel.we can maybe play around with the fact that cosx and sinx are derivates/integrals of each other
@rajdeepsingh26
Ай бұрын
I have a tip for you for better videos * better lighting
@BikeArea
24 күн бұрын
🙏
@melonking9752
Ай бұрын
Before watching it, my answer is 2/3
@RuthvenMurgatroyd
Ай бұрын
In the spirit of the intro meme 0:03 I like to think about it like this: dx² = 2xdx ∴ ∫xdx² = ∫x(2xdx) = 2∫x²dx = ⅔x³ + C. hence when the integral is evaluated from 0 to 1 it equals ⅔. 19:19 sin(x) is differentiable everywhere on the real number line so it's differentiable on that interval. dsin(x) = cos(x)dx ∴ ∫cos(x)dsin(x) = ∫cos(x)[cos(x)dx] = ∫cos²(x)dx = ½x + ¼sin(2x) + C. hence when the integral is evaluated from 0 to 1 it equals ½ + ¼sin(2) ≈ 0.7273243567.
@cheddastacker
Ай бұрын
Looking absolutely yoked man 🔥
@loganhagendoorn6327
Ай бұрын
love your content, thats such a smart way to solve this!
@theangledsaxon6765
Ай бұрын
Why haven’t your vids been recommended in so long?? Missing papa flammy!
@epicperson9961
Ай бұрын
This is how I did it: Let x² = t, Such that dx²/dt = 1, Hence dx² = dt. As a result the integral §xdx² becomes §√tdt which equals 2/3 √t³ + c Of which you can re-sub to obtain 2x³/3 + c. Substituting bounds, you then get 2/3.
@thekingofgindio
Ай бұрын
I'm afraid to ask why your right bicep seems larger than your left one
@PapaFlammy69
Ай бұрын
Masturbation I guess.
@thekingofgindio
Ай бұрын
@@PapaFlammy69 😎
@celestindupilon2773
Ай бұрын
@@PapaFlammy69 Aber PappaFlammy, du solltest doch wissen: 99, 100, Handwechsel!
@lithunoisan
Ай бұрын
@@PapaFlammy69what?
@wagsman9999
Ай бұрын
Look out, the guns are out to stay!
@jewgenijmoldawski3306
Ай бұрын
My first guess was: let y=x^2 and therefore x=y^1/2. Then integrate y^1/2 with respect dy. The result is of course also 2/3
@Djenzh
Ай бұрын
YESSSS PAPA FLAMMY IS BACK WITH SOME WEIRD INTEGRAL, LET'S FUCKING GOOOO!!!!!
@abhirupkundu2778
Ай бұрын
We got just use the substitution, x^2=t. x= root(t) assuming x is positive for this integral. So we got root(t)dt, = 2/3t^3/2= 2/3x^3, and applying the limits, we get 2/3
@laitinlok1
16 күн бұрын
Let u=x^2, du/dx= 2x, du= 2x dx , so the integral becomes x 2x dx, when u=0, x=0, when u=1, x=+-1.
@dragileinchen1485
29 күн бұрын
I really like to see the views on this vid. Hope this shows you, what your community really wants to see. I dont think more than 1% wants exponent rules.
@user-SK22-calc
Ай бұрын
we can compute dx^2 using the chain rule: dx^2=dx^2/dx*dx=(x^2)'*dx=2xdx int x dx^2=int 2x^2 dx=2/3x^3. after evaluation we get 2/3.
@robfielding8566
27 күн бұрын
Wow, what a complicated way to use the notation. This is how I do it: // my definition of integral, as a cancellation of S and d, where d is implicit diff operator S [d f] = f - f_0, d[f_0]=0 S (x d[x^2]) = S x*2*x dx = 2 S (x^2 dx) = 2/3 S d[x^3] = 2/3 x^3 - f_0 = 2/3 1^3 - 2/3 0^3 = 2/3
@crypticcrazy3672
16 күн бұрын
To equalize the subdivisions of the ordinate axis, I replaced x with SQRT(y) and dx^2 with dy, and the integral value of 2/3 pops out. (limits are unchanged) I have no idea if this presents any kind of general solution. I see others did the same.
@wjalp
Ай бұрын
Watched until the end! Also the assignment's answer is = 1/2+(sin(2))/4 :DD
@Kunal1255
Ай бұрын
So there are two branches of x for the equation y=x^2, hence I would expect that there are two possible solutions, namely +2/3 or -2/3, corresponding to each branch. Can you explain why we disregarded the other branch?
@7th_dwarf542
Ай бұрын
very clear and didactic 👏 thank you for this contribution
@GodzillaFreak
Ай бұрын
Wait am I coping? Can't we just index u = x^2? We would get int(0-1) u^(1/2)du Which solves to (2/3)u^(3/2)](0-1) And if we put back in x^2 for you we get (2/3)x^3](0-1) Which seems identical to the expected result. That seems to simple though so please tell me where I'm just completely wrong.
@thesuhasvasishta
21 күн бұрын
no you are absolutely correct,
@GodzillaFreak
20 күн бұрын
@@thesuhasvasishta Oh, that's great :D
@topquark22
Ай бұрын
In the Riemann integral, the partitions are equidistant. The point here is, they need not be.
@tomholroyd7519
Ай бұрын
The story of the Grothendieck prime 57 (see it on Wikipedia y'all) more than makes up for the shirt. I would have made a new shirt
@tomholroyd7519
Ай бұрын
It's a bit like Parker's Magic Square of Squares, very famous
@Hussain-px3fc
Ай бұрын
Before continuing the video the shirt caught my attention, why is 57 the best prime?
@PapaFlammy69
Ай бұрын
Grothendieck prime
@Hussain-px3fc
Ай бұрын
Oh I see, I didn’t even notice that it wasn’t actually a prime until now 😅 and great video btw
@marcosmaldonado7890
Ай бұрын
nice riemann-stieltjes integral👍🏻👍🏻
@ready1fire1aim1
Ай бұрын
Information-Based Unification of Forces: a) Central Idea: All fundamental forces (gravity, electromagnetism, strong, weak) emerge from a single information field. b) Unified Force Equation: F = -∇(ℏc/l_P² · log(I/I₀)) Where I is the local information density and I₀ is a reference density. c) Implications: - Potential resolution of incompatibilities between quantum mechanics and general relativity - New approach to grand unification theories - Prediction of new particles or forces at extreme energy scales
@melonking9752
Ай бұрын
I think we could've just solve it by saying x²=y and x=√y and so ∫¹₀ x dx² will be ∫¹₀ √y dy and the result will be (2y^3/2)/3 |¹₀ and the result of it is 2/3.
@Awdcguk
Ай бұрын
Use the absolute value for that
@melonking9752
Ай бұрын
@@AwdcgukBecause of the square root? Also do you think my way makes sense?
@a.g.2653
29 күн бұрын
Reminds me of line integrals in vectorial calculus
@boranxiii
Ай бұрын
isn't d(x²) just 2xdx?
@jorgeperezmolina2235
20 күн бұрын
Shouldn't the x^2 be in parentesis? I mean, dx^2 looks like dx*dx, while d(x^2) would look like 2x*dx. After all, in a second order derivative, where the notation used has no parentesis, we kinda mean dx*dx. I mean the "denominator" of the derivative, btw.
@firozabegum4373
Ай бұрын
"Nonlinear Partition Scaling" explains it all
@als2cents679
Ай бұрын
I did it is a much simpler way Integral [x d(x^2)] x^2 going from 0 to 1 = Integral [x du] u going from 0 to 1, where u = x^2 du = 2 x dx and when u = 0, x = 0 and when u = 1, x = 1 for the limits of the definite integral So, Integral [ x du ] u going from 0 to 1 = Integral [ x (2 x dx) ] x going from 0 to 1 = 2 * Integral [ x^2 dx ] x going from 0 to 1 = 2 * [ x^3 / 3 ] from 0 to 1 = (2/3) * [ x^3 ] from 0 to 1 = (2/3) * (1^3 - 0^3) = (2/3) * (1 - 0) = (2/3) * (1) = 2/3
@koenth2359
Ай бұрын
xdx^2 = x•2xdx = 2x^2dx = d(2x^3/3) So the value of the given definite integral is 2/3 - 0 = 2/3.
@m.h.6470
29 күн бұрын
just substitute x² with y, then you get Integral from 0 to 1 of √y dy, which results in 2/3. Done.
@hellohello-tf9vc
Ай бұрын
just write x=(x2)^1/2 and it becoomes a pretty simple integral
@flutterwind7686
Ай бұрын
d g(x) looks eerily similar to d g(x) / dx so the result seems kinda obvious in that way. d g(x) / dx = g'(x) can be rearranged to d g(x) = g'(x) dx
@jebarijihed
Ай бұрын
hey great video ! Is it possble so to do the integral of 1/dx ?
We should get the same value for the integral if it's just a different partition of the x-axis.
@appleducky5234
Ай бұрын
What would you do if the limits of integration don't match up? For all the examples you do dx^2 and dsin(x) the limits, 0 and 1 when plugged in result in new limits sin(1) = 1 and (1)^2 = 1 and similar for zero. For the Thumbnail integral the limits 0 to 2 don't match up and sinx never reaches 2. Since this trick operates on similar principals to U-Substitution wouldn't we need to change the limits of integration, and for the thumbnail example also split the integral into two pieces since sinx is not 1to1 from 0 to 2.
@Nerdwithoutglasses
23 күн бұрын
To those who say "why don't we use d(x^2)=2xdx or let u=x^2": surely you didn't read the title. Think twice or you will be r/wooooshed. This is not a place to show how good you are at "your calculus". Flammable Maths makes serious mathematic jokes, pay some respect.
@royalefighter0159
Ай бұрын
Here is the proof of the second way of calculating the integral: dg(x)/dx=g'(x) dg(x)=g'(x)dx ==> int{f(x) dg(x)} = \int{f(x) g'(x) dx} QED (Quit your Engineering Degree).
@emilleonardelli4047
Ай бұрын
17:35 I'm not sure and maybe I'm wrong, but isn't there a theorem that for a complex function a derivative exist? Does that mean complex functions are always integrateble that way?
@poyrazpekcan6635
Ай бұрын
I tried by saying dx^2 = du then integrated both sides however that gives me a + c (tho it conveniently gives the right answer for c = 0)
@wjalp
Ай бұрын
I really enjoyed this video! Keep it up! :DD
@bahaloicperrial8964
Ай бұрын
I didn't use this method. I instead use u=x**0.5, and i differiented both sides, and I applied it. It gave me 2/3
@alali2885
Ай бұрын
much much simpler would it be to just take d(x^2) = 2x*dx, then we would get 2*int(x^2 dx)|(0->1), and then basically (2*(x^3 /3))|(0->1) which is equal to 2/3...
@txikitofandango
Ай бұрын
Okay, this makes sense. I thought you were going to integrate over (dx)^2, not d(x^2), which would really be crazy
@Phaust94
Ай бұрын
Let's do Lebesgue-Stiltjes now
@wsollers1
Ай бұрын
I'm a simple man I see a feynman video and I slap the like button like it was a bongo
@dougr.2398
24 күн бұрын
dx ^ 2 = 2x dx
@dougr.2398
13 күн бұрын
Notation is unclear is dx^2 interpreted as (dx)^2 or d(x^2)??? Both in the video and my comment!!!!
@dougr.2398
13 күн бұрын
If the former is intended, then an additional integral sign is needed, so the latter is assumed to be the intent and correct
@Thalpathy6709
20 күн бұрын
We can use integral of u,v rule get same result
@thaianotran3315
Ай бұрын
isn't d x^2 = 2x dx? so the integral becomes the integral from 0 to 1 of 2x^2 dx
@tmlen845
Ай бұрын
Can't you just set X = x^2 (so x = √X), and then calculate the normal integral of √X d X, with X from 0 to √1? At least for this example, it seems to give the same result 2/3.
@broucho
18 күн бұрын
I don't get what you do at 11:52, in the parenthesis i think i squared disappear and i get 2i -1 but you write i(2i -i), and of course over n3
@MarcusPereiraRJ
Ай бұрын
But if g'(x) is a notation for dg(x)/dx, isn't it obviously deductible that dg(x) = g'(x).dx? Not trolling, really: what is so spectacular about that?
Is this the motivation for the 'd' operator for the exterior derivative?
@juliank.3522
Ай бұрын
How did you get [ 2*i - i ] in the left bottom corner of the first page? Thx
@williamangelogonzales148
Ай бұрын
A reimann-stieltjes integral!
@sobhhi
Ай бұрын
Gnarly farmer’s tan bro
@ViewtifulSam
Ай бұрын
I have one question. If the intuition is that we're dividing the segment according to the function g rather than linearly, why do we still take x_i to be i times delta x?
@sumdumbmick
Ай бұрын
it actually applies in all cases. it's merely that the standard case results in f'(x) being the multiplicative identity, so it's not necessary to be aware of it: if f(x) = x then f'(x) = 1 ∫ g(x) df(x) = ∫ g(x) f'(x) dx = 1 ∫ g(x) dx = ∫ g(x) dx
@reetomghosh2611
Ай бұрын
Arent integrals usually defined as integral over a to b of vdu? The logic is interesting but these dont seem anything out of the ordinary ?
@AmlanSarkar-wr2pr
Ай бұрын
Papa Flammy make a video on ISI (Indian Statistical Institute)entrance exam questions and on CMI (Chennai Mathematical Institute) entrance exam questions.These are pure math and statistics research institutes and the question level of these institutes are even higher than Jee advance.They are challenging problems you will surely like them.I guarantee you.😊😊😊
@evanwilliams7376
Ай бұрын
Imagine saying this only works for real functions and then going crazy with the i's.
@rubenvegas7926
Ай бұрын
Integral(dx)=x Thus, integral(dx^2)=x^2 So integral(x•dx^2)=x^3 Youre welcome Edit: I was trying to be a smartass before I watched the video. I had no idea this was going to be vaguely valid
@ninireak7325
Ай бұрын
That is ONE solution, if dx² means d(x²). But if dx² would mean (dx)² the solution is 0, because the volume under a surface of infinitesimal thickness tends to zero.
@nerdygeek8947
Ай бұрын
Flammy try integral(0 to 1) (x*d[x]) ;[x] is the greatest integral function
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