How to do two (or more) integrals with just one

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: https://www.seevid.ir/fa/w/gifQWtTWqEY The sequel to this video about fractional calculus: https://www.seevid.ir/fa/w/2dwQUUDt5Is A really nice video that derives the gamma function from scratch: https://www.seevid.ir/fa/w/v_HeaeUUOnc =Chapters= https://www.seevid.ir/fa/w/jNpKKDekS6k - Intro https://www.seevid.ir/fa/w/jNpKKDekS6k - Why Compress Integrals? https://www.seevid.ir/fa/w/jNpKKDekS6k - Analyzing the Problem https://www.seevid.ir/fa/w/jNpKKDekS6k - Visualizing a 2-Fold Integral https://www.seevid.ir/fa/w/jNpKKDekS6k - Deriving the Formula https://www.seevid.ir/fa/w/jNpKKDekS6k - Testing the Formula https://www.seevid.ir/fa/w/jNpKKDekS6k - How Is This Not Impossible? https://www.seevid.ir/fa/w/jNpKKDekS6k - Higher-Order Integrals https://www.seevid.ir/fa/w/jNpKKDekS6k - Application to Numerical Integrals https://www.seevid.ir/fa/w/jNpKKDekS6k - The Gamma Function =============================== 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/publication/331705202_Computation_of_Definite_Integral_Over_Repeated_Integral =============================== Want to support future videos? Become a patron at https://www.patreon.com/morphocular Thank you for your support! =============================== 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/morpholib

• #education
• #math
• #calculus
• #integration
• #integrals
• #repeated_integrals
• #cauchys_formula_for_repeated_integration
• #integral_compression