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Over the past years in math, I've seen many different methods for reparametrizing variables as a way to make a complex integral easier to deal with. Essentially we just take a function of some value and differentiate it with respect to a different variable, creating newer, simpler functions. One application of this includes u-substitution, where we let an arbitrary variable (in this case "u") represent a portion of our integral. For example, let's take this integral: | Over the past years in math, I've seen many different methods for reparametrizing variables as a way to make a complex integral easier to deal with. Essentially we just take a function of some value and differentiate it with respect to a different variable, creating newer, simpler functions. One application of this includes u-substitution, where we let an arbitrary variable (in this case "u") represent a portion of our integral. For example, let's take this integral: | ||
<center><math> \int {(sin{(x)})*(cos{(x)})} dx</math></center> | <center><math> \int {(sin{(x)})*(cos{(x)})} dx</math></center> | ||
| − | To solve this, we would simply let our new variable "u" equal <math>sin{(x)}</math> and differentiate both sides, resulting in an equation with <math> du = cos{(x)} dx </math>. We can then proceed to use this as a substitution for dx, changing our integral to <math> \int {sin{(u) | + | To solve this, we would simply let our new variable "u" equal <math>sin{(x)}</math> and differentiate both sides, resulting in an equation with <math> du = cos{(x)} dx </math>. We can then proceed to use this as a substitution for dx, changing our integral to <math> \int {sin{(u)} du}</math> |
, and we even see parametrization appear in surface integrals. | , and we even see parametrization appear in surface integrals. | ||
Revision as of 17:21, 27 November 2020
A Review on Reparametrizing
Over the past years in math, I've seen many different methods for reparametrizing variables as a way to make a complex integral easier to deal with. Essentially we just take a function of some value and differentiate it with respect to a different variable, creating newer, simpler functions. One application of this includes u-substitution, where we let an arbitrary variable (in this case "u") represent a portion of our integral. For example, let's take this integral:
To solve this, we would simply let our new variable "u" equal $ sin{(x)} $ and differentiate both sides, resulting in an equation with $ du = cos{(x)} dx $. We can then proceed to use this as a substitution for dx, changing our integral to $ \int {sin{(u)} du} $
, and we even see parametrization appear in surface integrals.
Feynman's Technique to integration utilizes parametrization and a combination with other different mathematical properties in order to integrate an integral that is can't be integrated through normal processes like u-substitution or integration by parts. It primarily focuses on setting a function equal to an integral, and then differentiating the function to get an integral that is easier to work with. A simple example would be: (1/2)*xe^(x^2) instead of just e^(x^2).
