Integrating by Weierstrass Substitution (visual proof)

In this video, we discuss the integration technique known as the tangent half-angle substitution, the Weierstrass substitution, or universal trigonometric substitution. This technique can be applied to find integrals for which other techniques might not work. We use it to find the integral of csc(x), which has another technique, and for 1/(1+sin(x)+cos(x)). Can you think of other integrals where this technique might work?
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This animation is based on a visual proof by Sidney H. Kung from the December 2001 issue of Mathematics Magazine (www.jstor.org/stable/2691035 - page 393).
#manim #math #mathvideo #mathshorts #calculus #triangles #animation #theorem #pww #proofwithoutwords #visualproof #proof #sinefunction #sums #pww​ #sine​ ​ #proof​ #algebra #trigonometry #mathematics​ #mathvideo​ #mtbos #integral #cosine #integrationtechniques #integration #halfangle #tangent #trigsubstitution
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[@jakobthomsen1595]

Very good visualization of a very elegant mathematical technique!

[@MathVisualProofs]

Glad you liked it! Thanks!

[@leif1075]

​​@MathVisualProofs WHAT wouldn't you agree I don't SEE ANYONE thinking of this substitution,no matter how smart theybare..wouldn't you agree? It's random or contrived and out of nowhere..Hope to hear from you.

[@bjoernschermbach3957]

I've been trying to memorise the identities for a while and now I can derive them reasonably quickly, that's so helpful!

[@MathVisualProofs]

Glad it helps!

[@catmacopter8545]

I learned about this substitution recently but this really explains the why! Thank you for your videos :)

[@MathVisualProofs]

Glad it was helpful!

[@EvilSandwich]

What's insane about this is that this is also virtually identical to how you can prove the Sum/Difference Formulas for Sine and Cosine.

[@MathVisualProofs]

Yes. I have that version in the works. If you check my channel you will see how many diagrams get repurposed for different results. This one shows up a few times. 😀

[@EvilSandwich]

@@MathVisualProofs Thats the coolest thing about math. So many seemingly unrelated things just keep unexpectedly tying together in weird and magical ways.

[@leif1075]

​@@EvilSandwichuea but math an also be reslly dumb and contrived and infuriating, all due respect..and isn't this an example of that..this just comes from contrivance not from intelligence right?

[@samueldeandrade8535]

​@@leif1075 what are you talking about?

[@EvilSandwich]

​​​@@leif1075Math, at it's core, is a tool used to examine the world in a more easily digestible way. And the streamlining of observation that it enables can sometimes reveal connections to other topics that weren't always apparent with all that messy reality in the way. So it's less a contrivance, and more just adding some clarity. Akin to finding a connection between two separate cultures that you didn't notice before you examined the grammar of both their languages.

[@glynnec2008]

Interesting video. Previously I've seen Weierstrass substitution explained using the inscribed angle (x/2) versus the central angle (x) on a unit circle.

[@shivanshnigam4015]

Alt method: multiply divide by (1-(sinx+cosx))

[@mekbebtamrat817]

Nice!

[@MathVisualProofs]

👍😀

[@adarshpradhan3554]

Bro that's amazing fr❤

[@MathVisualProofs]

Thanks!

[@yplayergames7934]

HOOOOOLYYYYY, this is aweeeesome, very nice job, now i understand it well

[@MathVisualProofs]

Glad it helped!

[@patrickt.4121]

awesome! thanks!

[@MathVisualProofs]

😀👍

[@Zana-iv9gp]

Visual proof helps alot to understand it ,Thanks!

[@MathVisualProofs]

😀👍

[@33arsenic75]

Amazing 🔥

[@MathVisualProofs]

👍😀

[@user-wj1qb3qu1y]

Amazing techniq

[@MathVisualProofs]

😀👍

[@alanthayer8797]

KEEP DOIN ya thang I WATCH ALL ya videos ! Thanks for Visuals 😊

[@MathVisualProofs]

Thanks for your support :)

[@Sunjidulsifat007]

Great visualization 😮

[@MathVisualProofs]

Thank you! Glad you liked it

[@shivanshnigam4015]

Example is any linear combination of sines and cosines can be integrated like this

[@williambusson3944]

you could try doing the intergrale of dx/((2-x)(1-x^2)^1/2) please

[@YoungPhysicistsClub1729]

if you scale the sides by z, won't the sides become lerger as ooposed to smaller? could you pls explain this to me

[@williammartin4416]

I am wondering the same thing

[@pranshukrishna5105]

why can't you solve by multiplying both sides 1 + sinx - cosx

[@thomasolson7447]

A:= (x,y,n) -> [[x,y], [-y, x]]^n B:=[1,0] B.A(1,z,2)=[1-z^2, 2*z] C:=(x,y,n) -> (x+i*y)^n C(1,z,2) = 1-z^2+i*2*z I call A the rotation matrix. Everyone tells me I'm wrong. I'm ok with being wrong. It is rotating by the angle [x, y] forms with [1,0]. The length is sqrt(x^2+y^2)^n. There is an ellipse that has a similar algebra. Both algebras are used in Mandelbrot Sets. I suppose it can also be called a translation matrix.

[@antoniodamianvargasmoreno]

Ok ...

[@user-ku9ky9mn4g]

Die Kunst!

[@MathVisualProofs]

Glad you liked it!

[@hydropage2855]

Hah here in India you should this vido to high school student and he will first be doing the laughter on you because then he will solve this in the next minutes because he was taught this technique when born. Ha

[@David-sj4fk]

This is incomprehensible.

[@hydropage2855]

@@David-sj4fk I’m making fun of them