- #1
mecattronics
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Homework Statement
The reduction formula is:
[tex] \int (1-x^2)^n dx = (1-x^2)^n x + 2n \int x^2(1-x^2)^{n-1} dx [/tex]
and the question is:
use this formula above how many times is necessary to prove:
[tex] \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)}[/tex]
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:
[tex] \int (1-x^2)^n dx = (1-x^2)^n x + 2n \int x^2(1-x^2)^{n-1} dx [/tex]
Integrating by parts:
[tex] =x(1-x^2)^n+2n \int x^2(1-x^2)^{n-1}dx [/tex]
let [tex] x^2 = -(1-x^2)+1 [/tex]
[tex]=x(1-x^2)^n-2n \int(1-x^2)^n dx + 2n \int (1-x^2)^{n-1}dx [/tex]
but I still don't know how to get there with this formula.
Any guidance would be appreciated.