The Trig Sub They Don't Want You to Know About

In this video we talk about the all-powerful Universal Trig Sub (also known as the Weierstrass Substitution)!!! We go over a few examples, and we will see the sneaky but simple geometric intuition behind this substitution.

The books I referenced in the video are:
- "Calculus and Analytic Geometry" by George B. Thomas
- "Calculus: Early Transcendentals" by Sullivan and Miranda

[aleksandarmanojlovic]

When I saw the title I immediately thought of exactly this substitution. Our teacher showed it to us, it is extremely powerful

[kutaykulbak]

the first time i heard about the weierstrass substitution was exactly like the intro of this video: a senior from my school barged into my dorm room and told me that he knew an op integration method that i won't hear from anyone else.

[merrickdodge]

Thank you all for the views, likes, and positive comments! I'm still new to the YouTube game, and seeing this level of interaction on this video was a pleasant surprise!

[ThatGuy-kfkc]

Me, currently doing my PhD in applied math, having never seen this substitution before

Adding this to my arsenal

[the_llaw]

Im not gonna lie that's definitely some fancy ass chalk. Must be really good to write with.

[cyberguardreal]

In the UK this is actually taught in great detail!!
Very cool sub

[gentlemandude]

Interesting. This substitution is widely taught in the UK, Australia and India, where it is called the "T-results" or "T formulae", but it seems that it is almost unheard of in the US. Very surprising! I learned this as a pre-calculus topic (in the UK). Very kind of you to share this with American students. You are correct. It is a game changer for integration. Bravo!

[danknowles]

As a UK A-level Maths/Further Maths teacher - this is now actually on the FP1 specification (A-level Further Maths) so lots of 17/18-year-old students will now have seen it! Very cool though, thanks for the vid :)

[pureatheistic]

I love that you keep your rubiks cube in a fun, but technically unfinished state. I do that as well, but I have it in crosses instead of the square form.

[toastyug]

this is so interesting! what a delightfully informative math video to pop up in my recommended. hope to see more of you in the future!

[liamturman]

Thanks so much, I feel much more confident in my trig subs

[spiralspark]

Woah, this video is so small yet so cool! Trying to teach myself calculus its hard to understand why substitutions are just allowed, but you explained in detail not only why this works, but also why it works generally with visual demonstrations. Keep it up!

[ginger]

wow.. I'm quite surprised not all teachers teach this substitution when it's this helpful. Definitely gonna come handy in my high school calculus. Thanks!

[hydropage]

I actually never really do that “out if nowhere” trick with integrating secant or cosecant unless I’m trying to be fast. I always wanted to know a much more reasonable way. For secant, I multiply by cos/cos, to cos/(1 - sin^2) which causes a u-sub with partial fractions. Way longer, but more reasonable

[edcify]

I'm not sure about calculus courses or books, but I'm pretty sure this substitution is very well known in classical analysis. Apart from exercises, the genuinely interesting thing to be said about it is its geometrical interpretation, and thus the meaning of the half angle formula for the tangent.

Haven't finished the video, but I do hope you mention it.

[rishitsharma]

Why is this video made so well, this is amazing

[ingiford]

I first this substitution in a number theory class when showing the 'rationalization' of a circle (ie rational points on the circle), and showing that you can get all the Pythagorean primitives, and was mentioned that it was also used in trig to prove identities and calculus as a substitution

[mb]

Hi, I had a look at your channel where you mostly discuss higher mathematics but this video in particular, I can relate to ! Would like more videos debunking some more of these weird tricks I had to mug up nearly a year ago for my competitive exams .

[DavidConnerCodeaholic]

i think i remember this from high school actually (i had BC calc), but it might have been from Calc 2. Unifying everything on the circle is what made all the strategies make sense.

[qwerasdliop]

In India we use this all the time, and it's not even given a name or anything, it was introduced to me in the middle of a calclus problem solving class. Dope muscles though and really well made video!