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 $$\geq$$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$$^{2}$$)$$^{n}$$ 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.