Consider the integral $$\int_{-1}^1\left(\sum_{j=1}^n\sqrt{j(2j+1)}P_j(x)\right)^2dx$$
How to evaluate the integral? Specifically, say, if $n=5$, then what would be the value? Which property of Legendre polynomials should I use here? Should I use completeness of Legendre polynomials here?
As pointed out in the comments by @Christoph, the answer is simple owing to the orthogonality of Legendre polynomials. We have: $$\left(\sum_{j=1}^n\sqrt{j(2j+1)}P_j(x)\right)^2=\sum_{j,k=1}^n\sqrt{j(2j+1)}P_j(x)\sum_{k=1}^n\sqrt{k(2k+1)}P_k(x)$$ Now, On integration, owing to orthogonality and the result $\int_{-1}^1P_n^2dx=\frac{2}{2n+1}$, we obtain, the required integral equal to $$2\sum_{j=1}^nj=n(n+1)$$