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} dxand 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} dxIntegrating 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.