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

[@jan-willemreens9010]

... 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

[@PrimeNewtons]

Thanks for your feedback. It means a lot.

[@jcmarketingblog375]

This channel has really come at the right time, thank you Newton🙏🙏

[@electricitysexenergy1509]

Which level

[@MohamedalieBangura-p1c]

​@@PrimeNewtonsHow are you doing sir? You have really inspired me to be a math major in my studies. But there's this website that say "this answer is the provided answer but it incorrect and the correct answer is tanx-secx+c" so what will be your take in this

[@KianParsian]

❤❤

[@asadrzayev3845]

You could also find sinx using the double angle formula: sinx = 2sin(x/2)cos(x/2). Thank you for the detailed explanation!

[@SunnyBoi.]

This is the most beautiful explanation of this theorem I've seen. I just recently learned about it in my Calc BC class and have been trying to understand how it works, but you just explained it perfectly. Thanks so much!

[@boydbanda1912]

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

[@PrimeNewtons]

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

[@BrianLewis-r5i]

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

[@PrimeNewtons]

Glad to hear that

[@BrianLewis-r5i]

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

[@kinyerajoel9731]

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

[@willwill3917]

you are a perfect teacher, moreover a perfect human

[@robread-jones3698]

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

[@Ni999]

Same here. 👍

[@prod1063]

you can also do this integration like this: ∫1/sinx+1= ∫1*(1-sinx)/(1+sinx)(1-sinx)dx=∫(1-sinx)/cos^2(x)dx = ∫1/cos^2(x)dx - ∫sinx/cos^2(x)dx= ∫sec^2(x) dx- ∫tanx*secx dx= tanx - secx + c. You can also check the above answer by differentiating the result. The answer will be 1/1+sinx Edit: but I like the answer in the video.

[@dusscode]

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]

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.

[@romanmuller997]

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

[@halimsemihozcan8526]

Thanks for this lesson . You are a good teacher .

[@senayurekyakan2248]

best teacher ever

[@georgesadler7830]

Professor Prime Newtons, thank you for the video. Calculus Two playlist on KZbin 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 KZbin TV with Professor Prime Newtons.

[@paulmatthewduffy]

Excellent refresher. Thank you.

[@ndzalamankwinika2999]

You are a good teacher, sir

[@mb-hv6kf]

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

[@utuberaj60]

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]

I'll give it a shot soon

[@mehakghous7884]

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 "❤❤❤❤"

[@_vblax]

life saver! great explanation, thank you!

[@Sapien1.618]

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

[@Lampiaosc2]

Very nice explanation.

[@arungosavi5698]

Brilliant way to solve sir

[@stevemwanza7521]

Clearly explained 👏👏 Thanks

[@mikedavis7636]

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]

That's really smart. Good job

[@hazwi]

i did the exact same process

[@estevaocachiliva3249]

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.

[@mattiollikainen8098]

Very good.

[@rawglang230]

Great Video👍🏻📈

[@levisim997]

Very helpful! ❤

[@fortunateonka]

Well explained sir❤

[@jan-willemreens9010]

... 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]

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

[@joseantonioandrade2808]

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

[@RaadoNoori-km5ej]

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

[@sssoup]

Excellent !

[@chappel9998]

Amazing video. You are so damn smart.

[@ThenSaidHeUntoThem]

Thank you! 😊

[@kurtecaranum3023]

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

[@lunaresting]

Thanks a lot man

[@UKPEINDANIELU.]

Fantastic

[@EE-Spectrum]

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

[@jef798]

Thank You!

[@Davics02]

Hi, ​when i multiply by conjugate (1-sinx), i get ans= [tanx-sec(x) ] at x= π/2 it diverges. While weierstrass substitution integral converges!!?

[@swarnabhamitra7233]

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

[@Bulbo_215]

Loved it

[@erms_234]

amazing

[@ramakrishnasalvankar5886]

What about multiply & divide by 1 - sin x and proceed ?

[@split9853]

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

[@KwaneleNgceboMamba]

thank you

[@domanicmarcus2176]

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

[@PrimeNewtons]

That's the substitution that works

[@Davics02]

Hi, ​when i multiply by conjugate (1-sinx), i get ans= [tanx-sec(x) ] at x= π/2 it diverges. While weierstrass substitution integral converges!!?

[@adebayoisraeladeshola]

Thanks sir

[@raymundorafaelito7532]

makasih king

[@adebayoisraeladeshola]

What if there are non linear function

[@36w]

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

[@TSR1942]

Try by multiplying the denominator and numerator with conugate.

[@d.yousefsobh7010]

Sir you also do it by substituting 1+sinx=

[@domanicmarcus2176]

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

[@deborahatobrah6827]

I like u boss

[@jjx-p6s]

l got you

[@chuckc3665]

this is a very bad method, you should trig identities

[@jusajiggynigga2524]

love your content bro