If I have shown that $\displaystyle \sum\limits_{k = 0}^\infty {{f_k}} $ is uniformly convergent on [0,1], where $\displaystyle {f_k}\left( x \right) = {\left( {1 - {x^2}} \right)^2}{x^k}$, how would I then go about showing that

$\displaystyle \int\limits_0^1 {\frac{{{{\left( {1 - {x^2}} \right)}^2}}}{{1 - x}}} {\rm{ dx}} = \sum\limits_{k = 1}^\infty {\frac{8}{{k\left( {k + 2} \right)\left( {k + 4} \right)}}} $?

All help appreciated!