I'm doing a problem involving legendre polynomials and I got stuck in this integral:
$$I_k=\int_{-1}^{1} x P_{2k+1}(x)dx $$
Update: Note that the function in the integral is even
If $k=0$, then $I_0=2/3$, but for the other values I don't know how to proceed
For integer $k>0$ I computed the integral in Wolfram(here is for k=1) and I obtained:
$$\int_{-1}^{1} x P_{2k+1}(x)dx=0 $$
Is this an identity for legendre polynomials? and if so, How can I prove it?