Reduction formula/Integration by parts problem

In summary, the conversation discusses a problem that asks for a reduction formula for a given integral. The problem includes a picture and a link to a website. The person is struggling with the problem, particularly with understanding why an extra r is not included in the denominator when integrating dv and the purpose of r^(n-1). Another person responds by explaining that they are using u = r^(n-1) and dv = re^(-ar^2)dr and performing integration by parts. They also clarify that v must equal (-1/2a)e^(-ar^2) in order to get dv back. The original person thanks them and expresses their appreciation for the website.
  • #1
Coffeepower
11
0

Homework Statement


The problem asks me to: Determine a reduction formula for the stated integral (See Picture and problem at:

http://garciarussellchem.angelfire.com/Photo/

Integration_by_parts.jpg

Please help me out with this problem. I don't understand why they do not include an extra r in the denominator when integrating dv. I also don't understand what purpose the r^(n-1) serves.

This is a bit confusing.

Thank you.
 
Last edited:
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  • #2
They are using [tex]u = r^{n-1}[/tex] and [tex]dv = re^{-ar^2}dr[/tex] then doing integration by parts with those...
 
  • #3
This means that v must equal"

v= (re^-(ar^2))/-2ar in order for us to get the dv back right?

THANK YOU FOR THE QUICK REPLY :) !
 
  • #4
No prob.

If you differentiate [tex]\frac{-1}{2a}e^{-ar^2}[/tex] you get [tex]re^{-ar^2}[/tex], so that's why [tex]v = \frac{-1}{2a}e^{-ar^2}[/tex]
 
  • #5
Thanks a bunch.
This site rocks.

(thank you)^(10)
 
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FAQ: Reduction formula/Integration by parts problem

What is a reduction formula?

A reduction formula is a mathematical formula used to simplify a more complex equation or expression into a simpler form. This is often used in integration problems, where the goal is to find the definite integral of a function.

How is a reduction formula used in integration?

In integration, a reduction formula is used to break down a complex integral into smaller integrals that are easier to solve. This is done by repeatedly applying the reduction formula until the integral can be solved using known integration techniques.

What is integration by parts?

Integration by parts is a method used to solve integrals of the form ∫u(x)v'(x)dx. It involves splitting the integral into two parts and using the product rule of differentiation to find the antiderivative of the function.

When should I use a reduction formula or integration by parts?

A reduction formula is typically used when the integral involves a power of x or a trigonometric function. Integration by parts is used when the integral involves a product of two functions. However, the choice ultimately depends on the complexity of the integral and the individual's preference.

What is the difference between a reduction formula and integration by parts?

The main difference between a reduction formula and integration by parts is their purpose. A reduction formula is used to simplify a complex integral into a simpler form, while integration by parts is used to solve integrals involving the product of two functions. However, both methods can be used to solve integration problems and may be interchangeable depending on the problem at hand.

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