Spherical Harmonics Normalization

1. Nov 29, 2011

mahblah

1. The problem statement, all variables and given/known data
I'm trying to solve

$$I_l = \int^{\pi}_{0} d \theta \sin (\theta) (\sin (\theta))^{2l}$$

2. Relevant equations

the book suggest:

$$I_l = \int^{+1}_{-1} du (1 - u^2)^l$$

3. The attempt at a solution

I think it's something related to Legendre polynomials

$$P_l (u) = \frac{(-1)^l}{2^l l!} \frac{d^l}{d u^l} (1- u^2)^l$$

but i don't know how to manage it... how it works?

thank u,
mahblah

2. Nov 29, 2011

dextercioby

This is amenable in terms of the Euler Beta function. Look up the definiton of Beta in terms of the integral of polynomials or sine/cosine and use it to express your integral in terms of Beta.