How to integrate 0 to 1 (1-x^2)^n

[mecattronics]

Homework Statement

The reduction formula is:
$$\int (1-x^2)^n dx = (1-x^2)^n x + 2n \int x^2(1-x^2)^{n-1} dx$$

and the question is:
use this formula above how many times is necessary to prove:

$$\int^{1}_{0} (1-x^2)^n dx = 2n \frac{2(n-1)}{3} \frac{2(n-2)}{5} ... \frac{4}{2n-3} \frac{2}{(2n-1)(2n+1)}$$
but I don't know how to get there.

Homework Equations

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The Attempt at a Solution

I tried to modify the reduction formula leaving it more recursive:

$$\int (1-x^2)^n dx = (1-x^2)^n x + 2n \int x^2(1-x^2)^{n-1} dx$$
Integrating by parts:
$$=x(1-x^2)^n+2n \int x^2(1-x^2)^{n-1}dx$$
let $$x^2 = -(1-x^2)+1$$
$$=x(1-x^2)^n-2n \int(1-x^2)^n dx + 2n \int (1-x^2)^{n-1}dx$$

but I still don't know how to get there with this formula.

Any guidance would be appreciated.

 

[hedipaldi]

use your last equality and induction.see attached.

 

[jedishrfu]

Since you have a definite integral perhaps you could tryparametric differentiation to get the answer.

The only reference I have for this is Prof Nearings Mathematical Methods pdf at:

http://www.physics.miami.edu/~nearing/mathmethods/

section 1.2 pg 4.

 

[SammyS]

Homework Statement

The reduction formula is:$$\int (1-x^2)^n dx = (1-x^2)^n x + 2n \int x^2(1-x^2)^{n-1} dx$$and the question is:
use this formula above how many times is necessary to prove:$$\int^{1}_{0} (1-x^2)^n dx = 2n \frac{2(n-1)}{3} \frac{2(n-2)}{5} ... \frac{4}{2n-3} \frac{2}{(2n-1)(2n+1)}$$but I don't know how to get there.

Homework Equations

The Attempt at a Solution

I tried to modify the reduction formula leaving it more recursive:$$\int (1-x^2)^n dx = (1-x^2)^n x + 2n \int x^2(1-x^2)^{n-1} dx$$Integrating by parts:
$$=x(1-x^2)^n+2n \int x^2(1-x^2)^{n-1}dx$$
let $$x^2 = -(1-x^2)+1$$
$$=x(1-x^2)^n-2n \int(1-x^2)^n dx + 2n \int (1-x^2)^{n-1}dx$$
but I still don't know how to get there with this formula.

Any guidance would be appreciated.

Your last line says that

$\displaystyle \int (1-x^2)^n dx=x(1-x^2)^n-2n \int(1-x^2)^n dx + 2n \int (1-x^2)^{n-1}dx\ .$

That has $\displaystyle \ \int (1-x^2)^n dx\ \$ on both sides of the equation.

Solve for $\displaystyle \ \int (1-x^2)^n dx\ .$

 

[mecattronics]

use your last equality and induction.see attached.

Hi hedipaldi, thanks for your help!

How did you get this formula?

 

[hedipaldi]

substitute the limits in your last equality.