Please make an illustration to me in proving of the following.
Problem: Assume that $ p_{n}(x)$ is a Legendre polynomials. I want to prove that $$ \int_{-1}^{+1}xp_{s}(x)p_{r}(x)dx=\delta_{r,s+1}\frac{2r}{(2r+1)(2r-1)}+\delta_{s,r+1}\frac{2s}{(2s+1)(2s-1)}. $$ For proof of this, I don't want to use the following recurrence formula $$ (n+1)p_{n+1}(x)=(2n+1)xp_{n}(x)-np_{n-1}(x). $$
What I have done:
First of all, I used Rodrigues' formula: $$ p_{n}(x)=\frac{1}{2^{n}n!}D^{n}((x^{2}-1)^{n}) $$ so $$ \int_{-1}^{+1}xp_{s}(x)p_{r}(x)dx=\int_{-1}^{+1}x(\frac{1}{2^{r}r!}D^{r}((x^{2}-1)^{r}))(\frac{1}{2^{s}s!}D^{s}((x^{2}-1)^{s}))dx$$ After this step, it seems that I have to use repeated integration by parts (because the proof of Rodrigues' formula is like that), but I can't use it.