The question: Let $(P_n), n \geq 0$, be the Legendre Polynomials, which is a total orthonormal system in real $L^{2} [-1, 1]$ with the inner product $\langle x,y\rangle = \int_{-1}^{1} x(t)y(t) dt$. Fix a $n \in \mathbb{N}$ and define $U(t) = (2n + 1)tP_{n}(t) -nP_{n-1}(t)$. Show that $U(t)$ is orthogonal to $t^n$ and $t^{n-1}$ respectively.
My work: I was going to use Rodrigues' formula, which says
$$P_{n}(t) = \frac{1}{2^{n}n!} \frac{d^n}{dt^n} [(t^2-1)^n]$$
and I was going to use this fact that
$$\int_{-1}^{1} (t^2-1)^ndt = \frac{2^{2n +1} (n!)^2}{(-1)^n (2n+1)!}.$$
What I have is this:
$$\langle U(t), t^n\rangle = \int_{-1}^{1} [(2n+1)tP_{n}(t) - n P_{n-1}(t)]t^n dt$$ $$ = \int_{-1}^{1} (2n +1) P_{n}(t) t^{n+1} dt - \int_{-1}^{1} n P_{n-1}(t) t^n dt $$
$$ = (2n + 1) \int_{-1}^{1} P_{n}(t) t^{n+1} dt - n\int_{-1}^{1} P_{n-1}(t) t^n dt $$
$$ = \frac{(2n + 1)}{2^{n}n!} \int_{-1}^{1} \frac{d^n}{dt^n} [(t^2-1)^n]t^{n+1} dt - \frac{n}{2^{n}n!}\int_{-1}^{1} \frac{d^n}{dt^n} [(t^2-1)^n] t^n dt. $$
I am not sure what to do here at this point. Could I possibly use symmetry for the integral? The problem is that I am not sure if the functions in the integrals are even or odd. I know there is a definite integral of a product of functions property. However, one of the functions has to be positive. So I am not sure if that applies here. Are there any ways I can continue on with this problem? Thank you very much!!