Show that the wavefunction is normalized

[Von Neumann]

Problem:

Show that

$\Theta_{20}=\frac{\sqrt{10}}{4}(3cos^{2}\theta-1)$

is a normalized solution to

$\frac{1}{sin\theta}\frac{d}{d\theta}(sin\theta \frac{d\Theta}{d\theta})+[l(l+1)-\frac{m_{l}^{2}}{sin^{2}\theta}]\Theta=0$

Solution:

I know how to show it's a solution, but I'm stuck on showing it's normalized.

I know that in general, a normalized wavefunction obeys,

$\int^∞_{-∞}\mid \psi \mid^{2}dV=1$

So would this particular normalized angular wavefunction obey the following?

$\int^{\pi}_0\mid \Theta_{20}(\theta) \mid^{2}\sin\theta d\theta=1$

I'm sorry if this is very elementary, but we just started doing this type of thing in my modern physics class. We haven't used any complex methods to solve these, so I don't think this problem will involve any advanced operators or anything of that sort. Any suggestions?

I just need help setting the integral up.

 

[DrClaude]

So would this particular normalized angular wavefunction obey the following?

$\int^{\pi}_0\mid \Theta_{20}(\theta) \mid^{2}\sin\theta d\theta=1$

Exactly.