Legendre Polynomial Integration

[member 428835]

Homework Statement

Simplify $$\int_{-1}^1\left( (1-x^2)P_i''-2xP'_i+2P_i\right)P_j\,dx$$
where $P_i$ is the $i^{th}$ Legendre Polynomial, a function of $x$.

Homework Equations

The Attempt at a Solution

Integration by parts is likely useful?? Also I know the Legendre Polynomials are orthogonal on $[-1,1]$. Before simply trying integration by parts term-wise I was trying to see if the equi-dimensional equation $(1-x^2)P_i''-2xP'_i+2P_i$ could first be expressed in a more simple manner? Any thoughts?

 

[member 428835]

Just realized $(1-x^2)P_i''-2xP'_i = ((1-x^2)P'_i)'$, and obviously the last term $2P_i$ instantly vanishes when $i\neq j$. Then the above integral becomes $$(1-x^2)P'_iP_k|_{-1}^1-\int_{-1}^1(1-x^2)P'_iP'_k\,dx+\frac{2}{2i+1}\delta_{ik}$$ where I use the kronicker delta. Does this simplify further?

Edit: yes it does further simplify. Look at Legendre's DE. It's fairly straightforward from here on out.