Help to prove a reduction formula?

[raphile]

Homework Statement

Let I_n = \int^{1}_{-1} (1-x^{2})^{n} dx.

Use integration by parts to show that I_n = (\frac{2n}{2n+1}) I_n-1 for n \geq1.

(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^{2})^{n} and dv/dx = 1. Hence, du/dx = (1-x^{2})^{n-1}(-2x) and v=x. Using this I got I_n = n \int^{1}_{-1} 2x²(1-x^{2})^{n-1} dx. This didn't seem to be very helpful.

I then tried writing I_n as \int^{1}_{-1} (1+x)^{n}(1-x)^{n} dx, and letting u be (1+x)^{n} and dv/dx be (1-x)^{n}. I thought this was a pretty clever idea, but it didn't give me what I wanted. I got I_n = \frac{n}{n+1} \int^{1}_{-1} (1-x²)^n-1(1-x)² dx, which can be written equivalently as \frac{n}{n+1} \int^{1}_{-1} (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 = (\frac{n}{n+1})(\frac{n-1}{n+2}) \int^{1}_{-1} (1-x²)^n-2(1-x)⁴ dx, or equivalently (\frac{n}{n+1})(\frac{n-1}{n+2}) \int^{1}_{-1} (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 = \int^{1}_{-1} (\sqrt{1-x^2)²ⁿ dx, and letting u = (\sqrt{1-x^2})²ⁿ and dv/dx = 1. This gave me a slightly more interesting result of I_n = 2n \int^{1}_{-1} x² (\sqrt{1-x^2})^2n-2 dx , i.e. I_n = 2n \int^{1}_{-1} x² (1-x^2})^n-1 dx, but I wasn't sure what to do next.

Please help... what should I be doing?

 

[HallsofIvy]

I think it would make a lot more sense to let u= 1-x² and dv= (1- x²)^n-1dx.

 

[raphile]

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.