I integrated the quadratic formula

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Ultimate math for fun! Last year we differentiated the quadratic formula, so today we will integrate the quadratic formula! This is my first time integrating the quadratic formula and I was surprised that the answer turned out to be pretty nice! This is a super fun problem to challenge any Calculus 2 student. In the end, we will check if ChatGPT can integrate the quadratic formula for us.
Check out how to differentiate the quadratic formula: • I differentiated the q...
0:00 We will integrate the quadratic formula
7:48 ChatGPT vs integral of the quadratic formula
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Thank you all!

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28 июл 2024

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Комментарии : 531

@blackpenredpen 9 месяцев назад

Check out how to differentiate the quadratic formula: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-JEcE-wDRMCk.htmlsi=Uq1p1KYNj-Kpu5g2

@KluhlHloo 9 месяцев назад

Integrating the quadratic equation = god mode! You sir are a badass!

@abishworpandit5200 9 месяцев назад

Well everything has physical meaning so what does it physically mean?

@KluhlHloo 9 месяцев назад

@@abishworpandit5200 🤔

@1Konu1Zoru 6 месяцев назад

dont rely on AI and dont ask AI to any math question.. it is very dumb in that area and still thinks that it knows the answer.. but in near future it will eventaully learn to do math and most likely beats any human being on earth :) wolfram is around for a long time and it is best tool to check results or solve something for the timebeing.

@BradleyG01 9 месяцев назад

If there’s one thing that ChatGPT tells us it’s that AI won’t be replacing math professors anytime soon.

@aashsyed1277 9 месяцев назад

Well that may not be true in the future....

@resurrectedpa 9 месяцев назад

@@aashsyed1277won't be soon 😅

@acuriousmind6217 9 месяцев назад

there's already multiple AI's online that can integrate any integratable function and give you the full meticulous steps idk if you count that as replacing math professors.

@littlegrass320 9 месяцев назад

@@acuriousmind6217 a math professor does a lot more than integrating functions, they deal with a lot more abstract concepts that don't have a set way to do them. Until those AI's can understand abstract concepts, math professors won't be going anywhere

@teelo12000 9 месяцев назад

ChatGPT write me a witty youtube comment reply.

@badamson 9 месяцев назад

If you change the last integral (1/(u^2-b^2)) with diff of two squares you can split it up with partial fractions and use natural log… although it turns out the end result is actually an identity for inverse hyperbolic tan

@Ninja20704 9 месяцев назад

They aren’t entirely the same because of the domain difference. tanh^-1 (x) is only valid for -1

@bobbyheffley4955 9 месяцев назад

@Ninja20704 if x>1 or x

@liamernst9626 9 месяцев назад

w pfp

@mrfarooqkhan8454 9 месяцев назад

@@Ninja20704bro can you explain why sometimes we get two different answers for the same integrant? How to decide which one is correct?

@Ninja20704 9 месяцев назад

@@mrfarooqkhan8454 Usually it is because the two different answers are actually just off by a constant, so they are essentially the same as far as the indefinite integral is concerned. One example is secx*tanx. In this case, the different answers we are getting is more of domain, but the functions have the same value for all x in the common domain. The integral of 1/(1-x^2) has 3 seemingly different answers: tanh^-1 (x), coth^-1 (x) and ln|(1-x)/(1+x)| tanh^-1 (x) = 1/2*ln[(1-x)/(1+x)], which is only good for -1

@CorrectHorseBatteryStaple472 9 месяцев назад

I love the grin at 6:03 with the inverse hyperbolic tangent. Math is fun and your subtle humour like this makes it fun.

@mrizwan4828 9 месяцев назад

😅

@skyethebi 9 месяцев назад

5:28 You can also use partial fractions for this integration u^2 - b^2 = (u - b)(u + b) 1/(u - b)(u + b) = 1/2b(u - b) - 1/2b(u + b) And that way you don’t end up with inverse hyperbolic trig

@xninja2369 9 месяцев назад

I was gonna write sameee

@perfectman3077 9 месяцев назад

Nerd

@xninja2369 9 месяцев назад

@@perfectman3077 stupid Sigma 10 year old kid , your place is not here go somewhere else dude 🤡 If you want to learn something than learn, if don't than get out ..

@joyboy69lffy 9 месяцев назад

so u will get 1=A(u-b) +B(u+b) so you cant find the value of A and B since you dont know neither the value of u nor b

@skyethebi 9 месяцев назад

@@joyboy69lffy My bad. You can still use partial fractions since b is constant with respect to the integration but you need to divide by b. I updated my answer so it should be right now.

@ahmedalhomaide4416 9 месяцев назад

Bro, I just graduated from high school but I can't ignore your video titles... Great as USUAL!

@blackpenredpen 9 месяцев назад

Thanks!!

@ethancasillas7325 9 месяцев назад

There is absolutely no reason for me to have watched this, I'm not in -or ever going to take- any calculus class where I have to integrate the quadratic formula. Yet here I am, loving every second of this video :)

@blackpenredpen 9 месяцев назад

Thank you!

@jamesdcarroll1 9 месяцев назад

Its been 30 years since I took this in college and I can still fall asleep halfway through.

@Alkis05 9 месяцев назад

Who said you would forget how to do the things you learned to do in your freshman calculus course?

@abyzdoof8821 9 месяцев назад

i love how we can just see all of those dry erase markers on standby in the bottom right. video’s fantastic as always, keep up the great work!

@acuriousmind6217 9 месяцев назад

A much more interesting approach is differentiating each quadratic formula with respect to each variable (a, b, c) and observing how each variable contributes to the fluctuation and the identification of the sweet spot for every variable that results in a more diverse set of solutions.

@dekippiesip 9 месяцев назад

Sure, a lot of cool things could be done with this. Take the solution set as a function with 3 variables, include complex numbers if you like, and look for all sorts of patterns. I'm thinking of stuff like maximizing the distance between 2 solutions for all vectors (a,b,c) of fixed length and shit like that. It's a heavily unexplored topic!

@MathNerd1729 9 месяцев назад

The last part reminded me of when I asked 2 different AI models (not ChatGPT) about whether they can prove Euler's Sum of Powers Conjecture . . . both of them had factual inaccuracies in their responses and one of them even claimed that it's currently unknown

@acuriousmind6217 9 месяцев назад

its the same as asking google a long question... at the end of the day its trained on pattern recognition you have to de-personify the ai so to speak when approaching it with complicated questions

@decaydjk8922 9 месяцев назад

All they are is token prediction systems, just a more advanced version of the predictive text in your phone. They can't "do" math or anything so no, that's not surprising

@abuabdullaahiwaaaishatah8235 9 месяцев назад

Lol

@lolerie 9 месяцев назад

Because only GPT 4 is AGI. Above human level.

@sunnohh 9 месяцев назад

Yeah ai is dogshit at medium hard or slightly complex

@gamingbeast5755 9 месяцев назад

Math Professors are the real JOD people on Earth....

@dayzimlich 9 месяцев назад

Great stuff, bprp! I love your videos. You teach but you do not talk down to the audience and you bring a catchy sense of humor and fun. Here's to your continued success!

@blackpenredpen 9 месяцев назад

Thank you very much for your nice comment!

@TranquilSeaOfMath 9 месяцев назад

Love the video. Easy to follow; well explained. I also like how you show the technology failure at the end.

@DatBoi_TheGudBIAS 9 месяцев назад

ive figured out a long time (by curiosity) dat chatgpt absolutely sucks with math problems. it can fail in something as simple as a quadratic equation, let alone a full integral

@anonymousaccount-27 9 месяцев назад

Its a fact that ChatGPT has cracked the toughest exams but does dumbest mistakes in new type of questions....

@elquesohombre9931 9 месяцев назад

Those exams aren’t usually logical ones, that and it will have many resources that allow it to “cheat” on those exams by pulling data from sites that have very identical questions/answers

@jok2000 9 месяцев назад

Just ask it to write a python program to symbolically integrate.

@SlipperyTeeth 9 месяцев назад

Me holding a stack of answers to every exam ever published online: It's a fact that I can answer the toughest exam questions, but am unable to apply any reasoning skills to "new types of questions".

@brookek3116 9 месяцев назад

@@elquesohombre9931Yep, I’d do pretty good on a lot of exams too if I was able to copy and paste answers from the internet

@Craznar 9 месяцев назад

Given it is only trained on language, that it even attempts to work out the answer is impressive.

@blackpenredpen 9 месяцев назад

Proving the quadratic formula while blindfolded -> ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-GWrqeFEAChY.htmlsi=AubYmfac-FODRJVj

@aaronyoung5871 9 месяцев назад

Damn

@thatomofolo452 9 месяцев назад

Yo 👋

@That_One_Guy... 9 месяцев назад

Can you differentiate and integrate the cubic formula ? Or even the quartic formula ?

@youssefbenmMorocco 9 месяцев назад

But why?

@mathwizardmathwizardmathwizard 9 месяцев назад

@@youssefbenmMorocco Because it's fun.

@winghei10 9 месяцев назад

Teacher: Maybe there is a mistake. Please verify the answer by differentiate the answer.

@gachanimestudios8348 9 месяцев назад

Legit the first question i asked from that title was "with respect to what?"

@zelda12346 9 месяцев назад

"I integrated the quadratic formula" has the same energy as "I sawed this boat in half!". And since you differentiated this already, "I sawed ANOTHER boat in half!"

@pulkitgupta5367 9 месяцев назад

Its giving the same energy as that flex tape ad where he cuts a boat in half and then tapes it back together: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-0xzN6FM5x_E.html

@tuseroni6085 9 месяцев назад

that trick of adding in a -b and a +b blew my mind. it makes perfect sense in retrospect, -b+b=0 so if you have something that is also something + 0 and thus you can put anything in so long as it cancels out to 0...

@blackpenredpen 9 месяцев назад

Yes 😃

@MayurAvad 9 месяцев назад

Really enjoyed this one !

@blackpenredpen 9 месяцев назад

Glad to hear. Thanks!!

@jok2000 9 месяцев назад

If you want ChatGPT to do advanced stuff you tell it to write a python program that does what you want like plot a function, do a symbolic integration or even get it to create a rotating globe in OpenGL. I've done all of these things with it and more... a subway station mapper using google maps API and TKinter etc. Some, like the maps API require a bit if tweaking but not the integrator.

@seanoneill2098 9 месяцев назад

You are doing good things !!

@A-0II0Io 9 месяцев назад

😂😂😂 the last part of your amazing video was amazing. Wow...

@ivanrodionov9724 9 месяцев назад

Is there a physical interpretation to integrating the quadratic formula with respect to the polynomial? Does it tell us anything special?

@spearmintlatios9047 9 месяцев назад

If we were to create a graph with an a axis and an x axis, where x is vertical (where y usually is) and a is horizontal (where x usually is), this would tell us the area under the curve of the graph. x = (-b+sqrt(b^2-4ac))/2a in terms of b and c which are constants. That’s really it, it’s not too special. If you wanted to see what this graph would look like you could go on Desmos and substitute x for y and a for x, and then create sliders for B and C.

@ivanrodionov9724 9 месяцев назад

@@spearmintlatios9047 Well that it can be viewed as an an area under a curve is one interpretation, however this is not what I am asking. He is doing something much more fun here, see, the quadratic formula is a function in polynomial of 2 degree space which has 3 degrees of freedom, a,b,c. It therefore maps a unique polynomial represented by the point (a,b,c) in the space and maps it onto two values which are it's roots. Now, when he integrates the quadratic formula, with respect to a, this is the question on how it changes the mapping and what does it tell us now. I am 99% sure there is quite a bit more going on here.

@spearmintlatios9047 9 месяцев назад

@@ivanrodionov9724I kinda get what you’re saying but I’m rusty on calc 3 stuff tbh. If you want, you can represent the quadratic equation as a graph mapping the function of a, b, and c to a value of x. or f(a, b, c)=x. this would be a 4 dimensional graph which is somewhat beyond our imagination, but everything should still apply. Since we only integrated with respect to A here, B and C are still able to be changed within the integral. pretend that B is equal to 1 for now. Then we can have a 3D graph of f(a, c) = x. In this case, we have a the a and c axis representing our domain, and an x axis representing our range. So for any point on the (a, c) plane, the output of this function is some “height” in the x axis creating the point (a, c, x). If this is the case, and we assume B is constant, then this integral represents the area summation of the height from the x output to the flat ac plane from a given start and end point a0 and a1, So it would still just be a 2D area found from an infinitesimal slice of a 3D graph. It is cool that we could move this slice by changing our bounds of a0, a1, and C. You could imagine changing these bounds as moving and stretching a tall piece of paper through the 3D space of the function, and calculating the area of that paper if you slice off all negative values and everything above the x points. To calculate the volume of the shape this graph makes, we would have to integrate by another bounds as well, a double integral. But the same logic works as well when we include B again. The integral would simply be a slice in a 4D shape that has X as the output. Unfortunately, we can’t just set X to 0 and have a graphical interpretation of 0 = integral (g(a, b, c))da. I don’t think there actually any representation there? If we wanted to find an integral that works here we would probably have to move some terms around before hand and treat it as a differential equation. Correct me if I’m wrong

@spearmintlatios9047 9 месяцев назад

@@ivanrodionov9724 after reading what you’re saying again I see you’re talking about something entirely different. Is there any tool that could create a mapping similar to what you’re talking about? I think what I was discussing might relate when you consider he only integrated using a plus and not a minus

@minseok8726 9 месяцев назад

@@spearmintlatios9047any applications in Engineering could you think of? Aerodynamics maybe

@mynamejeff69302 9 месяцев назад

Last part of integration with u^2-b^2 can be done with writing it as( u+b)(u-b) and then matching the numerator to get 1/2b^2 ln(u-b)/(u+b)

@panhandlejake6200 9 месяцев назад

So definitely an academically interesting exercise. Is there any insight about quadratics and their solutions that can be learned from this result?

@dhruvaraju5870 9 месяцев назад

No!

@granthqweqw5244 9 месяцев назад

Would love to see triple integration with respect to all variables there!

@SomeoneCommenting 8 месяцев назад

It would be interesting to compare the integrals of da, db, and dc for the same curve parameters to see what happens.

@thbb1 9 месяцев назад

Beautiful, but can we find a geometric interpretation of this integral? Or at least use it to gain some insights on the regions where the determinant is positive or negative as a function of the first coefficient of the trinome?

@gileadedetogni9054 9 месяцев назад

Maybe f(a)=all the formula

@Jordan-zk2wd 9 месяцев назад

Someone correct me if I'm wrong, but utilizing the intuition of the mean value theorem I think this means: if you vary a from some value a0 to a1, then the mean value of x which solves ax^2+bx+c=0 will be [f(a1)-f(a0)]/[a1-a0], where f(a)=-(b/2)ln|a|+sqrt(b^2-4ac)-(1/b)tanh^-1(sqrt((b^2-4ac)/b)* . I tried running it on desmos, I'm tired so I might have did this wrong, but here's an example of what I mean: from a=1 to a=2, the mean value of one of the roots of ax^2+3x+5=0 is -2.66324ish. I did have to throw in a negative sign at the end to make it all work, which means I'm made some small error along the way, but visual check of a few examples seem to confirm that it does work. * Important caveat, this will be the average value of one of the roots as you very a, the other one would have a slightly different definition of f(x) for the choice of the negative root of b^2-4ac, and thus give you a different average value corresponding to that root. @blackpenredpen ?

@OleJoe 9 месяцев назад

This is wild, man!

@perspicacity89 9 месяцев назад

Fantastic video, thank you.

@gamingbeast5755 9 месяцев назад

Nice Explanation Sir...

@muralidharrangaswamy9643 9 месяцев назад

I think you can also expand the denominator of the last integral into partial fractions and then express the result as a difference of two logarithms

@99chartered 9 месяцев назад

Thanks so much, it was fun!

@holyshit922 9 месяцев назад

6:09 If u is in interval (-1,1) but if u is outside this interval you have b/u as argument of inverse hyperbolic tangent

@guliyevshahriyar 9 месяцев назад

Very good lesson, thanks.

@disgracedmilo 9 месяцев назад

the sequel we needed

@alvaroarizacaro3451 9 месяцев назад

Muchas gracias. Muy bonita esta integral.

@Protactinium91 9 месяцев назад

I’m looking forward to learning hyperbolic tangent!

@arsalmathacademy 9 месяцев назад

I am always fan of this great person, brilliant brain

@arsalmathacademy 9 месяцев назад

@@English_shahriar1 Thanks

@madbrad6282 9 месяцев назад

That was fun to watch.

@xjk.- 9 месяцев назад

can we multiply both the num and denm with sqrt(b² - 4ac) when we split the integral -b/2a da +... and see what will happen?

@Krish-su4oh 9 месяцев назад

Last part was very meaningful 😂😂❤

@merixan9322 9 месяцев назад

I literally just watched the differentiating video of the quadratic formula yesterday

@NonTwinBrothers 9 месяцев назад

Im glad I woke up at 3am to enjoy this 😂

@adrienanderson7439 9 месяцев назад

Another one that's fun is the limit as a goes to 0, It gives an interesting result that makes sense when you think about it. If you really want a challenge you could do the limit of the cubic formula as the x^3 coefficient approaches 0 to see if you can get the quadratic formula

@euler1 9 месяцев назад

I did that like 2 month ago and thought it was very nice how the limit works out at the end. I don't know what you mean with the coefficient of x^3, what equation will you apply the limit to?

@adrienanderson7439 9 месяцев назад

@@euler1 there is a cubic formula but it is kind if complicated

@adrienanderson7439 9 месяцев назад

@@euler1 The solutions to ax^3+bx^2+cx+d=0 is given by the following: let u=b^2-3ac, let v=2b^3-9abc+27a^2d, let w=((v+-(v^2-4u^3)^(1/2))/2)^(1/3) where we can use any root for each cube or square root. Then x=(-1/(3a))(b+w+u/w) gives the roots

@euler1 9 месяцев назад

@@adrienanderson7439 It's the first I see this thanks for the explanation, I am curious to try it out but I think I will just look up a proof for it as I don't have that much time xd

@adrienanderson7439 9 месяцев назад

@@euler1 Yeah honestly I haven't done it myself and I tried to today but it really gets into the weeds

@Noobthepro0 9 месяцев назад

Instead of using inverse hyperbolic function as the integral of 1\(u^2-b^2), you may also use ln{(u-b)/(u+b)}/2b. That's how GPT might have also got the answer in terms of natural log

@parzflash1619 9 месяцев назад

We got the integral but can you explain the solution graphically as integration with limits gives area and what would the area of this graph be

@cameronskellams5670 9 месяцев назад

Thats hectic! Love the definitive proof at thr very end that chatgpt clearly has discalculia 😂

@jok2000 9 месяцев назад

Just ask it to write a symbolic integrator in Python.

@the_physics_pro.sohamkakka7765 9 месяцев назад

Wouldn't it be better to do trigonometric substitution in the 2nd integral instead of substituting the root as an algebraic variable?

@Tasz_ 9 месяцев назад

Came for blackpenredpen got blackpenbluepenredpen what a day to be alive

@yigiteldek 9 месяцев назад

i retried asking the integral to gpt couple times and with some hand it eventually integrated the formual exactly as in the video eventually

@vaccino3359 7 месяцев назад

Applied maths: Useful things that would probably come handy in life Pure maths:

@michaelbaum6796 9 месяцев назад

Nice solution - great👍

@blackpenredpen 9 месяцев назад

Thank you!

@eliteteamkiller319 5 месяцев назад

The ending had my rolling.

@justanotherguy469 9 месяцев назад

I love your t-shirt.

@Jesuisunknown 9 месяцев назад

I don't understand a lot but I'm getting interested again to learn the basics of Calculus

@dr0g_Oakblood 9 месяцев назад

Last segment is a perfect example of ChatGPT just making stuff up lmao

@inverted2533 5 месяцев назад

you know what, im startin to like this guy

@alijhi 9 месяцев назад

Brilliant !

@promethius7820 9 месяцев назад

This integral is basically the unbounded interval is basically the terms for the "missle knows where it is" problem, isnt it?

@francaishaitam6708 2 месяца назад

for the inverse hyperbolic tan part,you've better to go with a partial fraction decoposition ,since we don't now the domain of integration , it's ok to go with the inverse hyperbolic tan only if a is from (-1,1)

@user-ek9vo2ub9b 9 месяцев назад

So what can this be applied to specifically? Curious.

@fredrickfred4621 5 месяцев назад

You can also use trig sub to avoid hyperbolic tan

@tatoon34 9 месяцев назад

Basically integration is getting the area under the curve or easier formula to understand and so what is the area under the curve graphically or does it make it easier to realize after doing integration?

@patrickford7582 9 месяцев назад

I have a question, for u^2 -b^2 what if you replaced the constant b with say ib so that (ib)^2 was -b^2, then substituted it back it? That is use u^2 + (ib)^2 for u^2 -b^2.

@General12th 9 месяцев назад

Hi Dr. Pen! Very cool!

@blackpenredpen 9 месяцев назад

thanks!

@GuilhermeSilva-or8ud 5 месяцев назад

amo suas questões 🫶 um abraço do Brasil

@benberlowitz6381 9 месяцев назад

Please do a video where you triple integrate with respect to all 3 im genuinely intrigued what the result would be (is it even possible??)

@DiggOlive 9 месяцев назад

What does this look like if you plot it in 3D?

@AdaDenali 9 месяцев назад

I am struggling to interpret what the area under this curve would represent.

@AdakGamer 9 месяцев назад

Sir we can instead use 1/2u . ln[x-u]/[x+u] + c for the integration of 1/u^2 -b^2 du

@tomasbeltran04050 9 месяцев назад

It's been a while since I last saw one of your videos

@romanbykov5922 9 месяцев назад

very interesting video, blackpenredpenbluepen! :)

@blackpenredpen 9 месяцев назад

Thank you!!

@adarsh5997 9 месяцев назад

5:39 but the integrals of the form dx/(x²-a²)= 1/2a*ln(x-a)/(x+a) where ln is natural log

@said_qurbanov 9 месяцев назад

Bro will integrate my life

@maazqayyum693 8 месяцев назад

Could we not jist substitute u square with sec in u squared minus b squared?

@maxtieu9838 9 месяцев назад

Would there be any difference if it was the negative version?

@arnold-pdev 9 месяцев назад

Worth noting this function has a domain restricted to the interval 0 =< 4ac =< b^2. The second inequality comes from the discrimant (existence of roots). The first is a statement that either a and c are both negative or both positive. This is weird to me!

@jannegrey593 5 месяцев назад

Have you ever tried differentiating/integrating quadratic formula? Yes, basically as soon as I learned how to do basic derivatives from limits and noticing that circle/sphere calculations look like they have been integrated/differentiated using "power rule" (I think that's what it's called). 2*pi*r, pi*r^2 and 4/3 pi r^3 looked too much like coincidence. So I looked at quadratic formula (I also wondered back then what would happen if I did look for zeros when "delta" is "negative" - of course without imaginary numbers it is "harder" to do. Still doable in some cases and if you don't mind going a bit outside real numbers for rules). Sadly my knowledge was insufficient - had I tried it when I was more proficient it would likely help.

@enderoftime2530 9 месяцев назад

What does the integral tell us though. Is there an application for it?

@erichsu3325 9 месяцев назад

What if you integrate c^2=a^2+b^2+2abcos(C)?

@harshitmathur2390 6 месяцев назад

I wanted to ask if we can use this formula of integration 1 / x^2 - a^2 = 1/2a log(x-a/x+a) + C

@Saber09 5 месяцев назад

I am drunk right now but dude this video is absurd, it would never get to my mind this solution! Great! Have a good day :D

@JonRowlison 9 месяцев назад

So now that we have a formula... what does the integration of the quadratic formula represent (in layman terms)?

@Alkis05 9 месяцев назад

Wolframalpha still has a lock on the freshmen calculus student market.

@pradhyumnajadhav9137 9 месяцев назад

please do triple integration on this next

@composerlmythomorphic2635 9 месяцев назад

I got another (perhaps complicated solution: Suppose the integrand is x, where y=ax^2+bx+c. Then dy/dx=2ax+b, and dy/da=x^2. Hence da/dx=(2ax+b)/x^2. Then integral=int[(x)(2ax+b)/x^2]=int[2a+b/x]=2ax+b lnx +C

@ishanp-cp3pj 6 месяцев назад

Can we use the formula of Int 1/x^2-a^2 dx that is 1/2a Log|{x-a / x+a }| + c ?

@chakonleung8675 7 месяцев назад

What is the implication for this integration? Any significant findings about this?

@n4p3r0 9 месяцев назад

You should do it with respect to b and c next :3

@sless6928 9 месяцев назад

I'm just amazed you found 3 working whiteboard markers.

@EyeSooGuy 7 месяцев назад

Hey blackpenredpen … can you give us a calc THREE problem? I know that involves 3D shapes on a triple order graph. 😁

@teddy05p 9 месяцев назад

Great video !! I have an interesting problem for you:) Infinite series from k =2 of 1/(ln(k)^(ln(ln(k))) we are only allowed to use limit comparison or comparison. :)

@tomctutor 9 месяцев назад

lim (k=2 .. inf) f(k) = A + lim(n=27..inf) f(n) where A is some number and f(n) = 1/(ln(n)^(ln(ln(n))). Each term in the expansion of f(n) is now strictly positive, since ln(ln(27)) > ln(ln(e^3)) = ln(3)>ln(e) = 1, i.e. positive. So f(n) series terms are bounded above by 1, so only need to consider a monitonic decreasing series. Oh and also determine the leading constant A value.

@teddy05p 9 месяцев назад

@@tomctutor numerically this series diverges, so it is not bounded by 1.

@teddy05p 9 месяцев назад

@@tomctutor oh sorry misinterpreted your statement, it is, youre talking about the terms :)

@tomctutor 9 месяцев назад

@@teddy05p Yes I did not comment on wether or not the series converges. If alternating then it would converge, but it isn't alternating as I shown.