Show that Legendre polynomial satisfies $\int P_n(x)dx=\frac{1}{2^{n+1}}[P_{n+1}(x)-P_{n-1}(x)]$

Question

Show that Legendre polynomial satisfies $$\int P_n(x)dx=\frac{1}{2^{n+1}}[P_{n+1}(x)-P_{n-1}(x)]$$

Adding $$nP_n(x)=xP_n'(x)-P'_{n-1}(x)$$ and $$P'_{n+1}(x)=xP_n'(x)+(n+1)P_n(x)~,$$ we get

$$P_{n+1}'(x)+P_{n-1}'(x)=2xP_n'(x)+P_n(x)$$

I am supposed to integrate this and get the result, but that's not working. Please help!