Then express \(\displaystyle \alpha\) in terms of \(\displaystyle \beta\).

(Hint: Write \(\displaystyle (1 - x^2)^{10} = (1-x^2)^9(1-x^2)\) and use integration by parts.)

I feel dumb for not being able to do this...

This is what I've got:

\(\displaystyle

\begin{align*}

\alpha &= \int_0^1 (1-x^2)^9 (1-x^2) dx \\

&= \int_0^{2/3} (1-x^2)^9 d(x-\frac{x^3}{3}) \\

&= (1-x^2)^9(x-\frac{x^3}{3})|_0^{2/3} - \int_0^{2/3} (x-\frac{x^3}{3}) d(1-x^2)^9 \\

&= (1-x^2)^9(x-\frac{x^3}{3})|_0^{2/3} - \int_1^{\sqrt{1 - (2/3)^{1/9}}} 9(-2x)(1-x^2)^8 (x-\frac{x^3}{3}) dx

\end{align*}

\)

At this point, the problem has become such an intractable mess that I have no idea how to proceed (thoguh I've attempted several times). Is there something wrong I've done here? Thanks!