Nth order order integration help

  • Thread starter Jhenrique
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In summary: If you have a function ##f(x)## and you want to integrate it over a closed interval ##[a, b]##, then you can do it by using the following formula:$$\int_a^b f(x, y) dy$$This is often called the Cauchy-Schwarz formula.
  • #1
Jhenrique
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If exist differentiation until the nth order, so, why "no exist" integration until the nth order too? I never saw a quadruple or quintuple integral, and if exist, it's always with respect to different variables. Why not difine an integral so?

[tex]\int\int\int f(x)dx^3[/tex]
 
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  • #2
It is possible, I just don't know of any applications.
 
  • #3
Why bother?
Even the first integral is non-unique, what point would it be to gain a long, polynomial tail from your subsequent antidifferentiations??
 
  • #5
Jhenrique said:
If exist differentiation until the nth order, so, why "no exist" integration until the nth order too? I never saw a quadruple or quintuple integral, and if exist, it's always with respect to different variables. Why not difine an integral so?

[tex]\int\int\int f(x)dx^3[/tex]
I've never seen one written this way; i.e., with dx3. The usual way things are done is to have different variables of integration, something like this:
$$\int_a^b \int_c^d \int_e^f f(x, y, z) dz dy dx$$

or even like this:
$$\int_a^b \int_c^d ~...~\int_e^f f(x_1, x_2, ..., x_n) dx_1~ dx_2~...~ dx_n$$
Here we're integrating over a subset of Rn.
 
  • #6
It is possible to define a repeated integral, although the notation used is usually ##dx...dx## (n times) rather than ##dx^n##. The nth such repeated integral can be denoted ##f^{-n}(0)##.

And as long as the constants of integration at every step can be justifiably "ignored", e.g. ##f^{-1}(0) = ... = f^{-n}(0) = 0##, then it's easy to derive and prove a simple formula for the general repeated integral. See: http://mathworld.wolfram.com/RepeatedIntegral.html

You can derive it with integration by parts and prove the form by induction. Wiki also has something on this: http://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration

The general formula has an application in defining fractional integration.
 

1. What is nth order integration?

Nth order integration is a type of mathematical operation that involves finding the integral of a function that has been raised to the nth power. This is commonly used in physics and engineering to solve problems involving rates of change.

2. How is nth order integration different from regular integration?

The main difference between nth order integration and regular integration is that in nth order integration, the function being integrated is raised to a power before being integrated, whereas in regular integration, the function is simply integrated without any additional operations.

3. What are some common applications of nth order integration?

Nth order integration is commonly used in physics, engineering, and other scientific fields to solve problems involving rates of change. It can also be used in economics and finance to model growth and decay processes.

4. What are some tips for solving nth order integration problems?

When solving nth order integration problems, it is important to first understand the properties of the function being integrated and how they may affect the integration process. It can also be helpful to use substitution or integration by parts techniques to simplify the problem.

5. Are there any online resources available for help with nth order integration?

Yes, there are many online resources available for help with nth order integration, including tutorials, practice problems, and step-by-step guides. Some popular websites for this topic include Khan Academy, Wolfram Alpha, and Symbolab.

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