Integrating using t=tan(x/2) substitution - [The Weierstrass substitution]

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  • Опубліковано 7 лис 2024

КОМЕНТАРІ • 66

  • @jan-willemreens9010
    @jan-willemreens9010 Рік тому +41

    ... Newton, what I forgot to say is, keep doing your presentations on the blackboard with just a simple piece of chalk, your handwriting is excellent for this! Jan-W

  • @BrianLewis-r5i
    @BrianLewis-r5i 8 місяців тому +8

    Newton.. sincerely speaking you have helped me alot..🙏

    • @PrimeNewtons
      @PrimeNewtons  8 місяців тому

      Glad to hear that

    • @BrianLewis-r5i
      @BrianLewis-r5i 8 місяців тому

      @@PrimeNewtons thank u..I'm a Kenyan student

  • @boydbanda1912
    @boydbanda1912 Рік тому +13

    Big man you're doing it more than anyone. I like the way you Tutor us. Keep up the good work manh.

    • @PrimeNewtons
      @PrimeNewtons  Рік тому +1

      Thank you. Your comment sounds African 😀. Where are you from?

  • @dusscode
    @dusscode Рік тому +12

    You can actually solve it much more simply by multiplying the numerator and denominator by the conjugate of the denominator, to get ∫ (1-sin(x))/cos^2(x) dx.

    • @Koolasuchus8
      @Koolasuchus8 10 місяців тому +2

      Yeah but this method is very very useful for integrals that are impossible to figure out using other methods without already knowing the answer and working back. Its hard to explain but the "t method" is a very useful tool that some rusty person could use to solve integrals he is unprepared for or cant see the shortcut. Put simply its inductive not deductive. This is an easy example of the t rule.

  • @willwill3917
    @willwill3917 6 місяців тому +3

    you are a perfect teacher, moreover a perfect human

  • @kinyerajoel9731
    @kinyerajoel9731 3 місяці тому +1

    I am a teacher, but whenever I watch your video, everything is just fine to teach

  • @halimsemihozcan8526
    @halimsemihozcan8526 5 місяців тому +1

    Thanks for this lesson . You are a good teacher .

  • @robread-jones3698
    @robread-jones3698 Рік тому +14

    Thanks Newton, that was so good. I really enjoyed that.
    Keep up the great videos.
    👍

    • @Ni999
      @Ni999 Рік тому +2

      Same here. 👍

  • @ndzalamankwinika2999
    @ndzalamankwinika2999 Рік тому

    You are a good teacher, sir

  • @paulmatthewduffy
    @paulmatthewduffy 10 місяців тому +1

    Excellent refresher. Thank you.

  • @romanmuller997
    @romanmuller997 4 місяці тому

    Wow, this is a great video. You have such an excitement inducing voice. You're really getting the beauty of maths across.

  • @georgesadler7830
    @georgesadler7830 Рік тому +2

    Professor Prime Newtons, thank you for the video. Calculus Two playlist on UA-cam does not cover this topic. This topic is called special substitution, which is part of Techniques of Integration in Calculus Two. This is an error free video/lecture on UA-cam TV with Professor Prime Newtons.

  • @arungosavi5698
    @arungosavi5698 9 місяців тому

    Brilliant way to solve sir

  • @senayurekyakan2248
    @senayurekyakan2248 7 місяців тому

    best teacher ever

  • @mb-hv6kf
    @mb-hv6kf 4 місяці тому

    Thank you for the nice example and exposition. Added another tool to the toolkit.

  • @utuberaj60
    @utuberaj60 Рік тому +4

    Nice way to get the answer.
    But Mr Newton, I did it WITHOUT any substitution.
    In fact all I did was multiply and divide by the conjugate of the DENOMINATOR ( 1 - Sin X) and there onwards ,it is quite simple really for a good Calc 1 or Calc 2 level student.
    Your method, though a nice U-sub, seems quite lengthy, if I may say, Mr Newton.
    To make this a little harder, why not try to solve the same Integral as a DEFINITE integral from 0 to pi/2? It is quite an interesting one indeed, trust me❤❤

    • @PrimeNewtons
      @PrimeNewtons  Рік тому

      I'll give it a shot soon

    • @mehakghous7884
      @mehakghous7884 11 місяців тому +1

      Well for your information "SIR", this method (sir newton's one) really saved me since i had to do the integration USING this method in my calculus paper. Although your method is right, i guess your knowledge has made you arrogant, and you are trying hard to prove your supremacy to everyone by spreading hatred on the internet, and i guess that's why you're not teaching here with thousands of subscribers. have a great day "❤❤❤❤"

  • @RaadoNoori-km5ej
    @RaadoNoori-km5ej 4 місяці тому

    That is great, but I have question could we use t= tanx
    Or we have to make angle x/2

  • @estevaocachiliva3249
    @estevaocachiliva3249 4 місяці тому

    Great video professor👏🏽
    I have two questions:
    1 - can I use this t substituion whenever I have a integral of cosine and sine? And if I have another trigonometric identity can I rewrite this identity in terms of sine or cossine or both to apply this substituion?
    2 - how can I apply this substituion in those cases I have in the answer a angle in radians summing the variable on the argument of some trigonometric identity. Like for exemplo how to apply t substituion on the integral of dx/ sen(x) + cos(x).
    The answer of this integral is 1/sqrt 2 that multiply ln( csc( x + pi/4) - cot( x + pi/ 4)
    How can I reach the same result with t substituion.

  • @_vblax
    @_vblax 7 місяців тому

    life saver! great explanation, thank you!

  • @mikedavis7636
    @mikedavis7636 Рік тому +2

    I multiplied the numerator and denominator by the conjugate, 1 - sin X, got 1 - sin x/1-sin²X, substituted Cos²X for 1-sin²X, split the fraction, took the integral and ended up with tan x - sec X + c

    • @theandrewadler
      @theandrewadler 10 місяців тому

      That's really smart. Good job

    • @hazwi
      @hazwi 10 місяців тому +1

      i did the exact same process

  • @jan-willemreens9010
    @jan-willemreens9010 Рік тому +4

    ... A good day to you Newton, Right out of one of my " old and trusted " little math notebooks regarding integrals the following solution path in short: Given INT(1/(1 + sin(x))dx [ multiply top and bottom of the integrand by (1 - sin(x)) ] --> INT((1 - sin(x))/(1 - sin^2(x)))dx = INT((1 - sin(x))/cos^2(x))dx = INT(1/cos^2(x))dx + INT(- sin(x)/cos^2(x))dx [ u = cos(x) --> du = - sin(x)dx ] = INT(sec^2(x))dx + INT(1/u^2)du = tan(x) + INT(u^-2)du [ applying the good old REVERSE power rule, remember Newton? (lol) ] = tan(x) - 1/u [ u = cos(x) ] = tan(x) - 1/cos(x) + C = tan(x) - sec(x) + C = (sin(x) - 1)/cos(x) + C ... etc etc ... I leave the outcome to everyone's preference ... Thank you too Newton for your great performance; I really mean this, an eye opener for me; isn't it called the Weierstrass method? A pleasant weekend to you, Jan-W

    • @PrimeNewtons
      @PrimeNewtons  Рік тому +1

      Yes! This is an alternative. I call it 'rationalization'. I just get scared with the integral of secx or sec²x or sec³x. They scare me. But certainly, in recent times I have used that until I found my old Engineering Math book by K. A. Stroud. Then it all came back to me. We are dealing with the cold and rains here. It's never been like this before. Now I appreciate sunshine 🌞. And thank you for helping with the name. Truly it's called the Weierstrass Substitution

  • @EE-Spectrum
    @EE-Spectrum 6 місяців тому

    Is there a relationship between t-substitution and the half-angle identities?

  • @StrangeQuark1.618
    @StrangeQuark1.618 2 місяці тому

    Thank you so much :D
    It’s such a great explanation

  • @fortunateonka
    @fortunateonka Рік тому

    Well explained sir❤

  • @stevemwanza7521
    @stevemwanza7521 Рік тому

    Clearly explained 👏👏 Thanks

  • @KwaneleNgceboMamba
    @KwaneleNgceboMamba 3 дні тому

    thank you

  • @UKPEINDANIELU.
    @UKPEINDANIELU. 3 місяці тому

    Fantastic

  • @joseantonioandrade2808
    @joseantonioandrade2808 5 місяців тому

    Newton you can also solve that integral using the conjugate of 1+sinx , that is multiplying up and the bottom by 1- sinx, and the final result is tanx-secx, just another way to do it. greetings

  • @sssoup
    @sssoup 5 місяців тому

    Excellent !

  • @kurtecaranum3023
    @kurtecaranum3023 5 місяців тому

    If you remember all your derivatives, the integral could be solved as
    (1 - sin x)/(cos^2 x) dx
    (sec^2 x - sec x tan x) dx
    tan x - sec x + C

  • @adebayoisraeladeshola
    @adebayoisraeladeshola 3 місяці тому

    Thanks sir

  • @ThenSaidHeUntoThem
    @ThenSaidHeUntoThem Рік тому +1

    Thank you! 😊

  • @erms_234
    @erms_234 6 місяців тому

    amazing

  • @levisim997
    @levisim997 Рік тому +1

    Very helpful! ❤

  • @swarnabhamitra7233
    @swarnabhamitra7233 9 місяців тому

    Just use sinx=(2tan(x/2)) /(1+tan^2(x/2) ) =(2tan(x/2)) /(sec^2(x/2)) it becomes simple by half and double angles relations

  • @adebayoisraeladeshola
    @adebayoisraeladeshola 3 місяці тому

    What if there are
    non linear function

  • @chappel9998
    @chappel9998 10 місяців тому

    Amazing video. You are so damn smart.

  • @Bulbo_215
    @Bulbo_215 9 місяців тому

    Loved it

  • @lunaresting
    @lunaresting Рік тому +1

    Thanks a lot man

  • @TSR1942
    @TSR1942 9 місяців тому

    Try by multiplying the denominator and numerator with conugate.

  • @domanicmarcus2176
    @domanicmarcus2176 10 місяців тому

    Why did you choose x over 2 and not just Theta?

    • @PrimeNewtons
      @PrimeNewtons  10 місяців тому

      That's the substitution that works

  • @split9853
    @split9853 6 місяців тому +1

    How come the day after ive had my calculus 2 exam this video gets recomended 😒

  • @36w
    @36w 28 днів тому

    i had the integral 1/sin(x) but nothing worked I tried for like an hour and then I figured out with some googling that apparently it only works with tan

  • @d.yousefsobh7010
    @d.yousefsobh7010 10 місяців тому

    Sir you also do it by substituting 1+sinx=

  • @jjx-p6s
    @jjx-p6s Місяць тому

    l got you

  • @deborahatobrah6827
    @deborahatobrah6827 7 місяців тому

    I like u boss

  • @domanicmarcus2176
    @domanicmarcus2176 Рік тому

    Can we go backward? We now that sin^2(x)+cos^2(x) =1. Can we sub sin^2(x)+cos^2(x) for 1 and then separate our fractions like this: sin^2(x)/(1+sin(x)) +cos^2(x)/(1+sin(x)) and then keep manipulating our algebra to solve the problem in the video? Please let me know if it is possible? Thank You

  • @chuckc3665
    @chuckc3665 8 місяців тому

    this is a very bad method, you should trig identities

  • @jusajiggynigga2524
    @jusajiggynigga2524 Рік тому +1

    love your content bro