Help to prove a reduction formula?

In summary, the conversation discusses the problem of finding the integral In = \int^{1}_{-1} (1-x^{2})^{n} dx using integration by parts. The person attempting the solution tries various methods, including letting u = (1-x^{2})^{n} and dv/dx = 1, and also writing In = \int^{1}_{-1} (1+x)^{n}(1-x)^{n} dx and using another integration by parts. However, they are unable to find a solution. Another suggestion is made to let u = 1-x^{2} and dv = (1-x^{2})^{n-1}dx, but this also leads to an unknown integral involving
  • #1
raphile
23
0

Homework Statement



Let In = [tex]\int[/tex][tex]^{1}_{-1}[/tex] (1-x[tex]^{2}[/tex])[tex]^{n}[/tex] dx.

Use integration by parts to show that In = ([tex]\frac{2n}{2n+1}[/tex]) In-1 for n [tex]\geq[/tex]1.

(The integral above is supposed to be between the limits -1 and 1... sorry I couldn't figure out how to make the limits appear properly.)

Homework Equations



Integration by parts.

The Attempt at a Solution



I have tried several different ways.

First of all I tried letting u=(1-x[tex]^{2}[/tex])[tex]^{n}[/tex] and dv/dx = 1. Hence, du/dx = (1-x[tex]^{2}[/tex])[tex]^{n-1}[/tex](-2x) and v=x. Using this I got In = n [tex]\int[/tex][tex]^{1}_{-1}[/tex] 2x2(1-x[tex]^{2}[/tex])[tex]^{n-1}[/tex] dx. This didn't seem to be very helpful.

I then tried writing In as [tex]\int[/tex][tex]^{1}_{-1}[/tex] (1+x)[tex]^{n}[/tex](1-x)[tex]^{n}[/tex] dx, and letting u be (1+x)[tex]^{n}[/tex] and dv/dx be (1-x)[tex]^{n}[/tex]. I thought this was a pretty clever idea, but it didn't give me what I wanted. I got In = [tex]\frac{n}{n+1}[/tex] [tex]\int[/tex][tex]^{1}_{-1}[/tex] (1-x2)n-1(1-x)2 dx, which can be written equivalently as [tex]\frac{n}{n+1}[/tex] [tex]\int[/tex][tex]^{1}_{-1}[/tex] (1-x)n+1(1+x)n-1 dx. I then tried doing another integration by parts, letting u=(1+x)n-1 and dv/dx=(1-x)n+1, and this gave me In = ([tex]\frac{n}{n+1}[/tex])([tex]\frac{n-1}{n+2}[/tex]) [tex]\int[/tex][tex]^{1}_{-1}[/tex] (1-x2)n-2(1-x)4 dx, or equivalently ([tex]\frac{n}{n+1}[/tex])([tex]\frac{n-1}{n+2}[/tex]) [tex]\int[/tex][tex]^{1}_{-1}[/tex] (1+x)n-2(1-x)n+2 dx , but I still didn't seem to be any closer.

The other thing I tried was writing In = [tex]\int[/tex][tex]^{1}_{-1}[/tex] ([tex]\sqrt{1-x^2[/tex])2n dx, and letting u = ([tex]\sqrt{1-x^2}[/tex])2n and dv/dx = 1. This gave me a slightly more interesting result of In = 2n [tex]\int[/tex][tex]^{1}_{-1}[/tex] x2 ([tex]\sqrt{1-x^2}[/tex])2n-2 dx , i.e. In = 2n [tex]\int[/tex][tex]^{1}_{-1}[/tex] x2 ([tex]1-x^2}[/tex])n-1 dx, but I wasn't sure what to do next.

Please help... what should I be doing?
 
Last edited:
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  • #2
I think it would make a lot more sense to let u= 1-x2 and dv= (1- x2)n-1dx.
 
  • #3
HallsofIvy said:
I think it would make a lot more sense to let u= 1-x2 and dv= (1- x2)n-1dx.


But then I would have to integrate (1- x2)n-1dx, and I don't know how to do that when n is unknown. Am I missing something? I tried to integrate it using MAPLE, and it gave me something useless to do with the hypergeometric function.
 

1. What is a reduction formula?

A reduction formula is a mathematical tool used to simplify complex expressions or equations into smaller, more manageable forms. It is often used in the context of integration or differentiation problems.

2. Why is it important to prove a reduction formula?

Proving a reduction formula is important because it provides a systematic way to solve a class of problems that would otherwise be difficult or impossible to solve. It also helps to establish the validity and applicability of the formula.

3. How do you prove a reduction formula?

To prove a reduction formula, you need to use mathematical induction. This involves showing that the formula holds for a specific value, and then using this assumption to prove that it holds for the next value. This process is repeated until all values in the problem are covered.

4. Can a reduction formula be used for any type of problem?

No, reduction formulas are specific to certain types of problems, such as integration or differentiation. They cannot be used for all types of mathematical problems.

5. What are the benefits of using a reduction formula?

Using a reduction formula can save time and effort in solving complex mathematical problems. It also provides a systematic approach that can be applied to a wide range of similar problems. Additionally, it helps to deepen understanding of the underlying concepts and principles involved.

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