Integration by Parts: Formula & Real/Non-Integer n

In summary, the formula for integration by parts is only valid for integer values of n. If n is not an integer, the formula needs to be adjusted using the function h(x) in order to correctly perform the integration.
  • #1
zetafunction
391
0
is the following formula of integration by parts

[tex] \int_{-\infty}^{\infty}dxf(x)D^{n}g(x) = (-1)^{n} \int_{-\infty}^{\infty}dxg(x)D^{n}f (x) [/tex]

valid for real or non-integer n? the problem i see here is the term [tex](-1)^{n} [/tex] , which may be not so well defined for non-integer 'n'
 
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  • #2
Integration by parts takes place in integer steps, so the formula you presented requires n to be an integer. If n is not an integer, you need to use h(x)=Dng(x) and carry out the parts integration in integer steps. You will never get the right hand side.
 

What is integration by parts?

Integration by parts is a method used in calculus to find the integral of a product of two functions. It is based on the product rule for differentiation and involves splitting the integrand into two parts, one of which can be easily integrated while the other is differentiated.

What is the formula for integration by parts?

The formula for integration by parts is ∫udv = uv - ∫vdu, where u and v are functions and dv and du are their differentials. This formula is derived from the product rule for differentiation.

How is integration by parts used with real and non-integer n values?

Integration by parts can be used with real and non-integer n values by using the Gamma function, which is an extension of the factorial function to real and complex numbers. The Gamma function allows for the integration by parts formula to be used for a wider range of functions and values.

What are some common applications of integration by parts?

Integration by parts is commonly used in physics, engineering, and other fields to solve problems involving integration of products of functions. It can also be used to find the antiderivative of certain functions and to evaluate improper integrals.

Are there any limitations to using integration by parts?

Yes, integration by parts can only be used for certain types of integrals and may not always result in a solution. It also requires choosing the right functions to differentiate and integrate, which can be challenging for more complex integrals. Additionally, it may not work for some non-continuous or discontinuous functions.

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