Is there a way to turn multiple, repeated integrals into just a single integral? Meaning, if you, say, wanted to find the second antiderivative of 6x, is there a way to compute it all in one step just using a single integral? Turns out there is! In fact, any number of repeated antiderivatives can be compressed into just a single integral expression. How is that possible? And what does that single integral expression look like?

My old video about Double Integrals:    • How to Set Up Double Integrals  

The sequel to this video about fractional calculus:    • What Lies Between a Function and Its ...  

A really nice video that derives the gamma function from scratch:
   • How to Take the Factorial of Any Number  


=Chapters=
0:00 - Intro
0:51 - Why Compress Integrals?
2:29 - Analyzing the Problem
3:46 - Visualizing a 2-Fold Integral
5:25 - Deriving the Formula
10:56 - Testing the Formula
12:14 - How Is This Not Impossible?
13:49 - Higher-Order Integrals
15:22 - Application to Numerical Integrals
16:25 - The Gamma Function


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For more on applying Cauchy's Formula to numerical integration, see this paper:

Tvrdá, Katarína & Minárová, Mária. (2018). "Computation of Definite Integral Over Repeated Integral." Tatra Mountains Mathematical Publications. 72. 141-154. 10.2478/tmmp-2018-0026.
https://www.researchgate.net/publicat...


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The animations in this video were mostly made with a homemade Python library called "Morpho".
I consider it a pretty amateurish tool, but if you want to play with it, you can find it here:
https://github.com/morpho-matters/mor...