Understand u-substitution, the idea!

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  • Опубліковано 15 гру 2024

КОМЕНТАРІ • 171

  • @srpenguinbr
    @srpenguinbr 7 років тому +514

    I remember the first time I watched a video from your channel. I barely knew how to differentiate functions. When you did the u-sub, I didn't understand a word you said because I was used to thinking dx was just a notation that could tell what I call variable. I'm much better in calculus now. Thanks for making me a calculus enthusiast, together with 3blue1brown.

    • @blackpenredpen
      @blackpenredpen  7 років тому +88

      This is amazing to hear!! Keep up the good work and one day you will be great!

    • @ankitaaarya
      @ankitaaarya 5 років тому +9

      @@blackpenredpen how dx is not just a notation

    • @birdboat5647
      @birdboat5647 4 роки тому +1

      @@ankitaaarya en.wikipedia.org/wiki/Differential_(infinitesimal)

    • @brodierussell74
      @brodierussell74 3 роки тому

      I guess im asking the wrong place but does any of you know a trick to log back into an instagram account??
      I was dumb lost the login password. I would love any assistance you can offer me!

    • @srpenguinbr
      @srpenguinbr 3 роки тому

      @@brodierussell74 if you cannot send a recovery email os SMS you've probably lost it

  • @rizz.boy47
    @rizz.boy47 9 місяців тому +46

    His voice makes it easier to pay attention for some reason

    • @DMPatches
      @DMPatches 11 днів тому +1

      Was going to comment this. He is so clear and concise

  • @guitardudee777
    @guitardudee777 7 років тому +256

    my calculus professor is scary good, but he's not good in communicating mathematical intuition. We are always wondering how this became that. You, sir, are the complete package. You communicate mathematical intuition well and can explain everything, at the same time, you are scary good.

    • @dmorgan0628
      @dmorgan0628 7 років тому +10

      I know what you mean I had a professor that knew every aspect of Calculus but would be annoyed by someone asking precalc questions like we should have all of algebra and trig mastered coming into his lecture.

    • @vimuth_04
      @vimuth_04 2 роки тому +2

      Wow its been 5 years how’s it going 😁

    • @dylanrelatable
      @dylanrelatable 2 роки тому +1

      @@vimuth_04 Hello 😅

  • @infinitymfg5397
    @infinitymfg5397 8 років тому +97

    This was as really good explanation.

  • @xsytrance
    @xsytrance 6 років тому +17

    This was sooooo good! You blew my mind when you showed the connection between the chain rule and u-substitution! I've always had a problem choosing a candidate for the u. Thank you very, very much for this explanation!!

  • @Anuradha-cc1hh
    @Anuradha-cc1hh 4 роки тому +7

    Bro, you're amazing! You actually know how to explain a concept. God bless you with money. You saving dreams and careers out here!

  • @dmorgan0628
    @dmorgan0628 7 років тому +85

    In two years you will have so many views and subscribers especially since you continue to make so much content. I bet your students love you. I think Professor Leonard has some competition now haha

    •  5 років тому

      Bear down!!!

    • @eseasoro7251
      @eseasoro7251 4 роки тому +8

      U were right

    • @kepler4192
      @kepler4192 2 роки тому

      You are very much correct

  • @drakeaske9784
    @drakeaske9784 3 роки тому +9

    Awesome teacher! I bet your students do very well after seeing how easily you simplify things and show the connections

  • @n_2619
    @n_2619 4 роки тому +2

    By far the best explanation of u sub! Today in class my teacher tried to teach us this method and to say she destroyed the whole idea would be an underestimate. A large part of my skills in mathematics i owe to you! What an amazing teacher, keep going!

  • @OliviaBenFranklin-jg7ou
    @OliviaBenFranklin-jg7ou Рік тому

    You're the least confusing calculus youtuber I've encountered! Thanks for these videos!

  • @barthennin6088
    @barthennin6088 3 роки тому +1

    I finally "intuitively" understand "u" substitution for integrating! ..In High school and university, I successfully used u-sub to integrate but it was 'mechanical' w/o really understanding what was really behind it...Never realized it was just the chain rule in reverse! Yay!

  • @TheSuntriber
    @TheSuntriber 4 роки тому +2

    Once again you succeed in explaining what my textbook doesn't! Thank you!

  • @theartofmusic05
    @theartofmusic05 4 роки тому +1

    Man appreciate you for a living I have seen 60 videos about the substition rule of integral and I didn't understand nothing until I saw your video thanks bro I am 15 now years old and I really understand them

  • @dexio85
    @dexio85 7 років тому +8

    I wish you were my math teacher in the university. Love your videos.

  • @machoslothman
    @machoslothman Рік тому +1

    great explanation. trying to review material before i start my calc 2 course. you helped me bring my 50 up to an 80 in calc 1 this semester, thank you so much!!

  • @armanchowdhury1415
    @armanchowdhury1415 Рік тому +6

    The best, most clear and concise explanation I have ever heard. thank you too much, you are a great teacher.

  • @tobiasjdcolvin
    @tobiasjdcolvin 6 місяців тому

    Wow this is the first U sub video that made it click for me, thank you!

  • @ramyhuber8392
    @ramyhuber8392 22 дні тому

    Wonderful, just watched for the first time. Clear, well presented, easy to follow. Plus I like how you use the marker...

  • @Keiserm5
    @Keiserm5 Рік тому +1

    Simple explanation. Thank you so very much. You have made me so much more confident in understanding and using u sub. You should be very proud of your work!

  • @ahmeraymen2006
    @ahmeraymen2006 5 років тому +1

    u rock the calculus. man you save my brain and energy to understand the two basic concept

  • @TheZmoliver
    @TheZmoliver Місяць тому

    You're the king, man. Thank you once again. I was just watching some of these as a refresher but they really help refine my instincts (if that makes any sense) to where this stuff just becomes more intuitive. Snap! Just like 2nd nature.

  • @Treegrower
    @Treegrower 8 років тому +17

    Very clear explanation, thank you.

  • @robertadam512
    @robertadam512 8 місяців тому +1

    such a great explanation, thank you very much!

  • @taijmohabeer4515
    @taijmohabeer4515 Рік тому +1

    oh my god, you are genuinely an incredible teacher. i am fresh out of my o levels and am planning to take further maths and this is incredible

  • @palmtrees9474
    @palmtrees9474 2 роки тому +1

    You are a genius man! Hats off!

  • @johnedwards887
    @johnedwards887 2 роки тому +1

    Thanks!

  • @Kart-sl2qq
    @Kart-sl2qq Рік тому

    Best Video to understand as a calculus student myself, great Job!

  • @ehatipo4598
    @ehatipo4598 4 роки тому +2

    Amazing! heading down to the next video.

  • @andreamontano2621
    @andreamontano2621 7 років тому +10

    excelente vídeo, muchas gracias, saludos desde México!!

  • @josierodriguez9278
    @josierodriguez9278 2 роки тому +1

    Suddenly everything makes sense ….. THANK U !!!!

  • @pwnd785
    @pwnd785 5 років тому +3

    The second one can also be done as f’x/fx

  • @maznurahman2422
    @maznurahman2422 2 роки тому

    I love you very very much.I am a physics undergrad student.Your videos are helping me a lot.

  • @locutus7855
    @locutus7855 4 роки тому

    excellent - the IDEA behind u - substitution. Bravo

  • @GreaseMonkey33
    @GreaseMonkey33 7 місяців тому

    Bro you just helped me so much, what an absolute unit my guy

  • @frankharr9466
    @frankharr9466 7 років тому +6

    I agree with my caculus teacher, the hardest part of calculus is all the algebra.
    That seems clear and logical.

    • @MrCmon113
      @MrCmon113 5 років тому +2

      Nah, the hardest part is finding useful upper and lower bounds and proving stuff like continuity that always require a trick.

    • @black_jack_meghav
      @black_jack_meghav 5 років тому +2

      The hardest part is overcoming the illusion of difficulty.

  • @mauricioconejo102
    @mauricioconejo102 2 роки тому +1

    Thank you! I was struggling understanding this subject!

  • @Infinitesap
    @Infinitesap 6 років тому +1

    I simply love your videos. Great explanations.

  • @StuartSimon
    @StuartSimon 3 роки тому

    I had so much calculus in high school so as to get as far as what was just before L’Hopital’s Rule. When I was shown how simply integration by parts was derived to undo the Product Rule, it suddenly became clear to me that u-substitution was used to undo the Chain Rule.

  • @jarikosonen4079
    @jarikosonen4079 4 роки тому +1

    This is probably one of the best methods (along with D.I. method) to make integration easier... but finding right 'u' can be problem sometimes.

  • @d7neu250
    @d7neu250 7 років тому +2

    I haven't taken Calculus for the whole semester ... I'm a math student but I've been lazy and didn't really do much in college.. But I'm surprised I actually guess every steps you take and know the answer .. That was a brain refreshing!!! ... Thank you too much and I hope I can get back on track!! ❤❤👍👍👍

    • @blackpenredpen
      @blackpenredpen  7 років тому +2

      That's great! I am glad to help

    • @MrCmon113
      @MrCmon113 5 років тому

      Oh man, you'll be fucked.
      One tip: Look up the most important inequalities and series. Calculus is all about pattern recognition.

  • @ysmxysmx3292
    @ysmxysmx3292 3 роки тому +1

    this is my goat guys❤️

  • @adamkangoroo8475
    @adamkangoroo8475 7 років тому +27

    u're great :D

  • @awya1467
    @awya1467 Рік тому

    Isolate the dx seems like a real nice trick!

  • @tejaschakranarayan2760
    @tejaschakranarayan2760 5 років тому

    You are a genius i often watch ur videos n guss what?? I understands very quickly.. 👌👌👍👍👍👍 I love ur videos a lot ... Cheers to u👍👈👈

  • @albob8458
    @albob8458 3 роки тому +1

    Wow, I instantly understood u-substitution. Very clear and very concise!

  • @Beeboysquared
    @Beeboysquared 7 років тому

    So in the 1/5x-2 example, and others like it, setting u equal to the whole expression gets rid of the constant term when you differentiate. That's useful!

  • @zidanutomo326
    @zidanutomo326 3 роки тому +3

    "Let me show you"
    Proceeds to show 'U'

  • @pigchopORIGINAL
    @pigchopORIGINAL 5 місяців тому

    Fantastic explanation.

  • @oh.smetanka
    @oh.smetanka Рік тому

    Superb explanation!

  • @vishnukaliugavarathan5459
    @vishnukaliugavarathan5459 11 місяців тому

    was looking at the textbook for couple hours and was not able to solve a single question because it didn't provide any useful information on how everything is related now I am able to do the homework without looking over any examples just through 1 of your videos.

  • @m3lng3
    @m3lng3 Рік тому

    multiplying by d/dx os inside for chain, dividing by d/dx of inside for u - sub

  • @elendor3428
    @elendor3428 2 роки тому

    Fantastic explanation

  • @ryzenandrewgarcia657
    @ryzenandrewgarcia657 3 роки тому +2

    How about in these terms: the Chain Rule is post-derivative while the u-substitution is the pre-derivative, following its connection.

  • @SinisterPrince
    @SinisterPrince 2 місяці тому

    Thank you, bro! This helped me a lot.

  • @rubensf7780
    @rubensf7780 9 місяців тому +1

    Helpful

  • @Headerman300
    @Headerman300 6 років тому +4

    What is the first integration technique? How many techniques exist? Great video bprp i love your work

    • @KevinJonesPandaas
      @KevinJonesPandaas 3 роки тому +1

      This is speculation, but it probably is the power rule of integration (for monomial, add +1 to degree and divide the term by the new power). It could also just be “know your basic derivatives”.
      As for the other question, I’m not sure. As many mathematicians say, “Differentiation is a tool, while Integration is an art.” There are the big strategies, like U-Sub, Trig-Sub, Integration by Parts, and Taylor Approximations, but solving an integral is like a map- the individual examines the routes and uses their knowledge to show the quickest path.

    • @rickdoesmath3945
      @rickdoesmath3945 3 роки тому

      The first technique is calculating the integral from the definition, and it actually works for all the elementary functions (like if you have monotonicity then it's an easy win)

  • @lucc6619
    @lucc6619 2 дні тому

    in highschool it was organic chemistry tutor and khan academy. In university it's blackpenredpen

  • @ebadrahman1290
    @ebadrahman1290 4 місяці тому

    simple and informative

  • @protocolwonder4558
    @protocolwonder4558 10 місяців тому

    Great explanation

  • @isi6402
    @isi6402 3 роки тому +1

    This help me in A level Exam

  • @isaacneilton3497
    @isaacneilton3497 7 місяців тому

    Adoro os vídeos desse cara

  • @benjaminknudson5997
    @benjaminknudson5997 7 років тому +23

    Isn't it?

  • @LepusCompositions
    @LepusCompositions 10 місяців тому

    very helpful video. thank you :DDD

  • @zabul4423
    @zabul4423 5 років тому +1

    u r best teacher. isn't you

  • @iternai3872
    @iternai3872 11 місяців тому +1

    For the first integral, when you divided by 4x^3, wouldn't that mean x can't equal to zero?

  • @rob6129
    @rob6129 4 роки тому

    Thanks for this video, now u-substitution seems less esoteric to me ^^

  • @doodelay
    @doodelay Рік тому

    Woooooow it's all becoming clear to me now. I did not see the connection to the chain rule.

  • @anonym4162
    @anonym4162 5 місяців тому +1

    0:40 Horse power

  • @ratnarajwora2674
    @ratnarajwora2674 6 років тому +2

    How to find the area under the curve x^4 + y^4 = 2xy

  • @stephenfreel2892
    @stephenfreel2892 4 роки тому +1

    I always thought multiplying dx/du was geometrically kind of like dilation of another functions integral such that it matches the integral of the original function

  • @YusufKhan786
    @YusufKhan786 6 років тому +1

    So when you integrating you actually multiplying by the dx at the end? Doesnt the dx at the end have nothing to do with the sum? Is it not just there to help identify the variable of integration?
    Please explain

    • @snillie
      @snillie 5 років тому +3

      Yeah, the dx does actually represent a quantity being multiplied! If you think of the integral as giving you the area under some graph, you can imagine approximating this area by adding up lots of rectangles side-by-side to each other with a certain width (which we can call dx, standing for change in x, since this is also the change in x of the horizontal position of each rectangle) and whose height just touches the graph of the function you're integrating. Then if you imagine letting dx approach 0, getting smaller and smaller, this rectangle approximation should get closer and closer to the true area, since you're chopping up the area into finer and finer rectangles. So what the dx in the integral truly represents is the behaviour when you let dx approach 0. If you want a clearer explanation of this with visuals, I'd highly recommend 3blue1brown's Essence of Calculus series. It'll help clear up a lot of "why" questions in calculus as well as just this one :D

  • @debopamsil6965
    @debopamsil6965 6 років тому +1

    Use the chen lu

  • @rohanvenkatesh9643
    @rohanvenkatesh9643 4 роки тому

    2:50 we U-sually

  • @jeena563
    @jeena563 4 роки тому

    Thank you so much to be my teacher

  • @adrianmacias9144
    @adrianmacias9144 5 років тому

    Where was this video when I needed to learn this before my chapter 5 exam 🤣

  • @zeyuanluo9707
    @zeyuanluo9707 6 років тому +5

    I get the technique... but I cannot visualise what is going on graphically when we do the substitution. When we integrate a function with respect to dx, we are breaking the area under the function into many pieces of width dx and finding the area of each piece and then take the limit of the sum as dx becomes 0. In this case dx is constant so this is pretty intuitive.
    But clearly du is not constant as it changes with x. For example in the first example, du=dx*4x^3, and we are integrating with respect to du. To me, this doesn't make sense because how do we integrate with respect to something that is not constant? Would appreciate if someone can give a visualisation of what is going on (graphically) when we integrate by substitution.

    • @robmarks6800
      @robmarks6800 5 років тому

      zeyuan luo how is du not constant? dx is constant, and the 4x^3 will cancel out with some of the integrand, for the substitution to result in some new function

    • @ILikeFeelingElectric
      @ILikeFeelingElectric 5 років тому +1

      zeyuan luo When you’re integrating, the function on the inside is the same. You’re just changing the form of the function, and changing what variable you are integrating in terms of. So, it’s the same area under the curve, but of a function that looks different than the original.

  • @Lerky
    @Lerky 4 роки тому +1

    this dude saves more grades than teamtrees plants trees

  • @vrowniediamond6202
    @vrowniediamond6202 7 років тому

    Just realized I have the same book... Well thanks Mr! :)

    • @zomisintu
      @zomisintu 7 років тому

      May I know which book is that?
      Thanks

  • @andregosteli2177
    @andregosteli2177 8 років тому

    Very nice!

  • @carvelbell181
    @carvelbell181 7 місяців тому

    excellent.

  • @lorenasiu3113
    @lorenasiu3113 8 місяців тому

    Thank you so much

  • @BlackZeus1990
    @BlackZeus1990 3 місяці тому

    Thank you

  • @whoppers7778
    @whoppers7778 5 років тому

    you are #1 !!!

  • @RadulovicDragan
    @RadulovicDragan 2 роки тому

    Bravo!

  • @ValidatingUsername
    @ValidatingUsername 8 місяців тому

    Man I am digging deep to rationalize this even though I get it from just a memorization standpoint but I have so many issues with it.
    The point of integrating or deviating a function is that the result is useful and equates to something. With that being said that value is equal to the operator denoted by the integral sign sandwiched by the dx.
    So there literally is a dx as soon as you try to integrate to obtain that value or d/dx when taking the derivative.
    When you u sub the x^4 you’re finding dx for that function the same way choosing 4x^3 would result in 12x^2 dx and subsequent dx = du/12x^2.
    But that wouldn’t be useful if you’re trying to change the base of the integration operator from dx to du because it would simplify to int{ x/3sec^2(x^4)du right?

  • @mohsinashfaq101
    @mohsinashfaq101 2 роки тому

    Great explanation but how do we know which one is 'U'????

  • @kubinn7989
    @kubinn7989 2 місяці тому

    nice vid so helpful

  • @gameliadeti2967
    @gameliadeti2967 2 роки тому

    Sir please why is it that, after differenting sec square, the du disappear?

  • @Anonim01089
    @Anonim01089 4 роки тому

    So cool!

  • @alexanderpoltzer8885
    @alexanderpoltzer8885 6 років тому +2

    But i am confused, why cant you substitute everything to u? I know that it doesnt work i was just wondering why.

    • @crosisbh1451
      @crosisbh1451 6 років тому

      because you'll still need to take the derivative of u, which could end up being a mess depending on the function.

    • @zeyuanluo9707
      @zeyuanluo9707 6 років тому

      You have to replace dx with the relevant notation first by expressing it in terms of du. To do so, you find dx/du and multiply both sides by du (treat dx and du as very small numbers). If your u is the original function in terms of x, then dx/du is the derivative of the inverse function of u. This is just as troublesome to find as the original equation.

  • @ngsbsad5502
    @ngsbsad5502 Рік тому

    Guys can someone tell any tips on when will i know if i need to use this coz im confuse when im solving with the basic integration especially if it's a hard problem with square roots

  • @exilitygamin3388
    @exilitygamin3388 7 місяців тому

    How to know when to use this method!

  • @polyblankas
    @polyblankas 10 місяців тому

    MUCH LOVE TY

  • @pulanridick7040
    @pulanridick7040 5 років тому

    thanks man!

  • @higherle-bi2yi
    @higherle-bi2yi Рік тому

    thx

  • @llennzo
    @llennzo 2 місяці тому

    x world to the u world. Got it!

  • @aashsyed1277
    @aashsyed1277 3 роки тому

    use the chen lu!

  • @shouu000
    @shouu000 4 роки тому

    How would you differentiate the function with the absolute value included?

    • @carultch
      @carultch Рік тому

      Unless it is a special case where the absolute value signs are ultimately irrelevant, the answer eventually becomes a piecewise function.
      For instance, d/dx |x^3| = piecewise 3*x^2 when x>=0, and -3*x^2 otherwise.
      By contrast, d/dx |x^2| is still 2*x, because the absolute value signs are redundant (at least for the real numbers), as the original function already is always positive.
      Another example is d/dx ln|x|. This one we KNOW is 1/x, which is valid for both negative x and positive x. But why? Initially, it may seem like a coincidence, that all it takes is absolute value signs to reconcile the integral of 1/x, as the integration operation cuts the domain in half. But what is really going on, is that the +C is arbitrary, and is different on both halves of the function. If you let the +C include an imaginary term, left of the origin, you'll see that ln(|x|) + C is really the full complex log, when the +C can change upon crossing the origin. You can take the log of a complex number, and it is ln|x| + 2*pi*k*i, where k is any integer.

  • @dxmady9500
    @dxmady9500 Рік тому

    "Are you doctor yet ?"
    blackpenredpen's dad.

  • @spudhead169
    @spudhead169 4 роки тому

    Why u though? Any particular reason to use u rather than another letter?

    • @carultch
      @carultch Рік тому

      There often is no particular reason why a specific letter is used in mathematics. It very likely is simply because it was the first letter that came to mind, that wasn't spoken-for, when the technique was coined. For instance, why do we call spatial directions, x, y, and z? Probably because that's the trio of letters that is least likely to stand for anything specific, so it is the de-facto choice of a variable in general.
      Since it's common that t is a variable of integration standing for time, they simply picked t's alphabet neighbor as a placeholder variable of an intermediate step within the integral.
      Some letter choices might appear to stand for something specific, but turn out to be completely coincidental. Like e standing for Euler's number. It isn't called e because it stands for exponential or Euler. It's just that Euler had a preference for picking vowels, and a was already spoken-for, so he picked the next vowel of the alphabet, when he coined his famous number.

    • @spudhead169
      @spudhead169 Рік тому

      @@carultchI guess, it just seems a poor choice. If you need to do a further substitution, the natural choice would be to use the next letter after u, namely v. But those two can easily be mistaken, especially with handwriting.