# Help to prove a reduction formula?

1. Dec 6, 2008

### raphile

1. The problem statement, all variables and given/known data

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

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

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

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

Last edited: Dec 6, 2008
2. Dec 6, 2008

### HallsofIvy

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

3. Dec 6, 2008

### raphile

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.