I want to do the following integral:
$$ I=\int_{-1}^{1} x^n P_{n}(x) \rm{d}x $$
WITHOUT using Rodrigues' formula. I'm required to use
$$ P_{n}(x) = \sum_{r=0}^{[n/2]} \frac{(-1)^r (2n-2r)!}{2^n r! (n-r)! (n-2 r)!} x^{n-2 r}. $$
Substituing $ P_{n}(x) $ into I, I got
$$ I = 2 \sum_{r=0}^{[n/2]} \frac{ (-1)^{r} (2n-2r)!}{2^n r! (n-r)!(n-2r)!(2n-2r+1)} $$
, but I don't know how I can proceed.