(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

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

2. Relevant equations

Integration by parts.

3. 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 I_{n}= n [tex]\int[/tex][tex]^{1}_{-1}[/tex] 2x^{2}(1-x[tex]^{2}[/tex])[tex]^{n-1}[/tex] dx. This didn't seem to be very helpful.

I then tried writing I_{n}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 I_{n}= [tex]\frac{n}{n+1}[/tex] [tex]\int[/tex][tex]^{1}_{-1}[/tex] (1-x^{2})^{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 I_{n}= ([tex]\frac{n}{n+1}[/tex])([tex]\frac{n-1}{n+2}[/tex]) [tex]\int[/tex][tex]^{1}_{-1}[/tex] (1-x^{2})^{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 I_{n}= [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 I_{n}= 2n [tex]\int[/tex][tex]^{1}_{-1}[/tex] x^{2}([tex]\sqrt{1-x^2}[/tex])^{2n-2}dx , i.e. I_{n}= 2n [tex]\int[/tex][tex]^{1}_{-1}[/tex] x^{2}([tex]1-x^2}[/tex])^{n-1}dx, but I wasn't sure what to do next.

Please help... what should I be doing?

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