How to Normalize Spherical Harmonics Using Euler Beta Function?

[mahblah]

Homework Statement

I'm trying to solve

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

Homework Equations

the book suggest:

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

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

 

[dextercioby]

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