TITLE: Show the Integration of Legendre Polynomials is 0
QUESTION [5 upvotes]: Show that for the Legendre polynomial $P_n$ with $n\neq0$,
$$\int_{-1}^{+1} P_n (x) dx =0$$
I put this polynomial in the Legendre equation then got stuck. Can you help me find out what to do next?

REPLY [9 votes]: You know that $$\int_{-1}^{1}P_{n}(x)P_{m}(x)dx=0$$
This is the orthogonality property. Now $P_{0}(x)=1$, and so you have the  following:
$$\int_{-1}^{1}P_{n}(x)dx=\int_{-1}^{1}1\times P_{n}(x)dx=\int_{-1}^{1}P_{0}(x)P_{n}(x)dx=0$$
This is true for $n\neq 0$