weierstrass substitution for integrations, intro

[@NGBigfield]

Erasing the board in an instant. you are a pro!

[@Koisheep]

I studied this method both in high school and college but now I'm told it has a name

[@aryanks2167]

its prolly easier to remember with a name

[@maalikserebryakov]

@@aryanks2167yep

[@CDChester]

I must say that most professors I meet absolutely hate this method. I have loved using it. I show the kids I tutor this method early on in Calculus (after teaching identities). It makes Calculus much easier in my opinion. As well I love how you were saying Double as Daaable. I say that too now and my students get a kick out of it. Overall, thanks for the video BPRP! #muchlove

[@Emily-fm7pt]

It's just so ugly though, I can see why your profs would hate it lmao

[@maalikserebryakov]

@@Emily-fm7pt cry . Can’t do simple algebra? And if you hate it so much then try the Rules of Bioche Firstly let the rational trig integrand = w(t) and do a couple of tests Test 1 - Positive phase shift invariance If w(pi+t) = w(t) Then u=tan(x) will ALWAYS be a good substitution Test 2 - Negative Phase Shift Invariance if w(pi - t) = w(t) then U= Sin(x) will always be a good sub Test 3 - Negative Argument invariance If w(-x) = w(x) then u=cos(x) is always a good substitution. 4 -If TWO invariances are shown, then u=cos(2t) is also always a good substitution If none of this applies, THEN use weierstrass This is is the most expedient and logical method of approaching a trigonometric rational structure. The reason it works is because TRIG functions are Periodic so all you have to do is exploit their phase shift invariances The fact trig is periodic SHOULD mean we get another insight into solving/integrating it right? Compared to non periodics like polynomials at least.

[@rkusuma6852]

damn, someone got burnt down here.

[@ageofkz]

extremely clear! Way clearer than the wiki article. Thanks for the video 👍

[@jesushernandez6944]

eres un excelente profesor de matemática y se te ve pasión que sientes por la matemática. Gracias por tan didáctico y grandioso canal.

[@ruskodudesko9679]

right off the bat - I love you mic I love you enthusiasm and I love your voice, I'm gonna watch lol - I hate math tutors that suck and you sir do not

[@trinoromo6619]

What kind of sorcery is this!? Thank you.

[@OptiInfo00]

I remember when my professor teach me this in class, in that moment I just thought he was some kind a genius; then when I did the demostration myself I feel like a genius, and now watching your video I remembered all that 👍

[@plaustrarius]

This is such a serious concept, it directly gives pythagorean triples and a rational parameterization of the circle basically given the original substitution (t = tan(x/2)) and some good trig skills. the dx part is clever for sure, thats more about functional analysis so definitely within the grasp of a precalc student, but I would love to see the original motivation Weierstrass had to find this method, or if its simply attributed to his name. I would guess because it simply gives a nice set of relationships with all of the variables, t, x, sinx, cosx, etc. and an analytic geometry sort of perspective on the problem, like a foothold where no relationships seemed to exist.

[@jimjam1948]

He never even used it in his worm. It was named after home but the earliest use of the substitution is in the work of the one and only Leonhard Euler

[@plaustrarius]

The one and only hahaha everything is named after him and even things that aren't he probably discovered or made famous XD Where is our next Euler?!

[@jimjam1948]

@@plaustrarius Ikr. Early kyler I need you to rise up and become our next Euler

[@nottoday3817]

I have an exam tomorrow. You might have just saved my life

[@mabindisarorisang]

Oh what the!! I get it now. Thanks so much, this was helpful

[@AndDiracisHisProphet]

Weierstraps :D Your videos are absolutely brilliant.

[@holyshit922]

Argument of tangent can be shifted by a constant value For example to calculate integral of secant using this substitution , we can shift argument of tangent by \frac{\pi}{4} to get less calculations I think that he should draw bisector of one of the acute angles Try to use this substitution with inverse trig substitution/

[@aggbak1]

Jacek Soplica LaTex does not work on youtube

[@ecehanvergiliel2672]

THANK YOU SO MUCH!!! It helped verrryyyy wellll :))

[@burkay9133]

nasıl geçti sınavın

[@llll3073]

رحم الله والديك

[@EpicMethGaming]

why is this so menacing wtf

[@vaibhav8090]

Nicely explained sir!

[@ssdd9911]

is there another version that involves hyperbolic functions?

[@robertolasagna1212]

great video. i love the mic as well

[@MikeB3542]

The "sneakiest substitution"!

[@holyshit922]

Calculations needed for Weierstrass substitution are comparable to the Euler substitutions You showed Weierstrass substitution but Euler substituton only on one example There are three Euler's substitutions but two of them (with leading coefficient and with real roots) will be enough to cover all integrals in the form R(x,sqrt(ax^2+bx+c))

[@maalikserebryakov]

Why ever use euler sub Just complete square

[@npola3713]

There is an identity: tan(x/2)=sqrt([1-cos(x)]/[1+cos(x)]), which makes it easier to draw a triangle rather than using double angle formula everytime you want to find sin(x), cos(x).

[@johnhwhittaker6005]

Skill unlocked

[@blackpenredpen]

John H Whittaker nice!

[@williamvlidel3497]

Hi sir, may I ask, if you treat tan(x/2)=t and then draw a right triangle, with 1 as the adjacent side. Won't it be valid only for quadrant I and IV since cosine is positive only in those quadrant. Or is tan(x/2)=t true for all quadrant, but why? since the adjacent side is only 1 and the only side that can be +/- is the opposite side with the variable t. Thanks!

[@015Fede]

William Vlidel this is just an intuition on how to get the other functions. The more rigorous way would be to apply known trig identities

[@ianmoseley9910]

If you have an indefinite integral you can choose to solve for first quadrant.

[@ernestschoenmakers8181]

Steve you're the new Harry Potter.

[@MathZoneKH]

Pretty good sir

[@lostwisdom8900]

Awesome! Is there a video in your channel where you explain the t = tanx formula too? Thanks for all your work.

[@ionlycommentwhenitsnecessa5906]

If tan(x/2) = t, wouldn't arctan(t) = x/2, instead of ctg(t) = x/2?

[@EarlyMonAF]

In our notation, tan¯¹(x) is arctan(x) and not ctg as you would expect. He did a recent video about the confusion that the notation causes so you are definitely not alone. The (F)¯¹ = arcF notation is not logical but we're stuck with it. Btw, ctg is cot for us.

[@erwinrojasarabia]

So beautiful video

[@bizarrapmusic]

i don't seem to understand how you formed the right triangle? like why did you make t the oposite side ?

[@Private_Username007]

Because tan x = opposite side/adjacent side. So t is the opposite side and 1 is the adjacent side

[@Virendersinghawanagood]

thanks from vs gujjar India

[@deepakm9342]

Clearly explained thank u !!!!

[@vlix123]

for cosx couldn’t you use difference of squares to end up with cosx=1-t

[@Rehan-ld1zw]

no you can't. (1+t)(1-t)(1+t^2); (1+t^2) is not equal to (1+t)^2

[@calimaulud5708]

My teacher told that there are actually 100+ formula for integral and I want to know and study them.

[@herberthubert6828]

amazing!! thank you!

[@user-yg97f5hfvh]

how about x/2 is over ㅠ/2 ?

[@domc3743]

this is the best thank you

[@genologic210]

it is every clear you are the best

[@user-mv4ix7jd8o]

What just happened on 3:31? I don't see the connection between sin(2*x/2) and 2sin(x/2)cos(x/2)

[@jomariraphaellmangahas1991]

Double Angle Identity

[@yannickescalera3231]

Super nice!

[@user-bz7ct3iu3v]

But why tan(x/2)?

[@fyrerayne8882]

Brilliant

[@suhaybseeman4699]

thanks teacher

[@miyamotomusashi4556]

If one watches this video carefully, they will see a very elegant method to find sin ( arctan(x) ) !

[@jamesnitsuga4985]

Are u singaporean?

[@9909faisal]

wow thank you man,just WOW

[@andreapaps]

I donno whats cooler the maths or the moustache :D

[@davidlue3082]

why not use 1-sin~2=cos~2 so cos= t`2-1/ 1+t~2 is not same to yours

[@matokurin]

great great great

[@holyshit922]

In fact we can draw other triangle to illustrate this substitution I created puzzle which shows substitutions which rationalize integrands in the form R(x,sqrt(ax^2+bx+c)) because it is two in one substitution (inverse trig substitution and shifted Weierstrass substitution)

[@noway2831]

weirdass substitution

[@mangojuic3e]

vielen dank meine kerle