I am wondering if there is a way to compute the following integral:
$\int_{\frac{\pi}{2}}^{\pi}\left[P_0(\cos\theta)P_k(\cos\theta)\sin(\theta)d\theta\right]$
I have very little experience working with Legendre Polynomials and I am not sure how to approach this situation. I have found formulas for calculating them over a half-interval, however, I am not sure how it changes when the scale factor: $\sin\theta$ is introduced.