Assuming the result

P^{'}_{n+1}(z)-P^{'}_{n-1}(z)=(2n+1)P_{n}(z)

I need to show that

\int^{1}_{-1}z^{n}P_{n}(z)dz={\frac{n}{2n+1}}\int^{1}_{-1}z^{n-1}P_{n-1}(z)dz

and then deduce that

\int^{1}_{-1}z^{n}P_{n}(z)dz={\frac{2^{n+1}(n!)^{2}}{(2n+1)!}  }

Any pointers on where to start with this?