Tough integrals with Legendre polynomial

Question

Does anybody here know how to integrate $\int_0^\pi P_n(\cos(x))\sin(x)\cos(x) dx$, $\int_0^\pi P_n(\cos(x))\sin^2(x) dx$, where $P_n$ is the n-th Legendre polynomial? They are actually extremely hard to do, as far as I see, but I pretty much need them.

Answer 1

Rewrite the integral as

$$-\int_0^{\pi} d(\cos{x}) \, P_n(\cos{x}) \cos{x} = \int_{-1}^1 dy \, y \, P_n(y)$$

By orthogonality, the integral on the right is zero unless $n=1$. Therefore,

$$\int_0^{\pi} dx \, P_n(\cos{x})\, \cos{x}\, \sin{x} = \begin{cases}\frac{2}{3} & n=1 \\ 0 & n \ne 1 \end{cases}$$

Answer 2

Hint: $P_1(\cos x)=\cos x$. The first integral is

$$-\int_0^\pi P_n(\cos x)P_1(\cos x) d\cos x=\int_{-1}^1 P_n(y)P_1(y)dy=... $$

All you need it to use the orthogonality property of Legendre polynomials.