Particle in Ring (Griffiths)

Consider a particle with mass $\mu$ constrained to move on a circle (radius $r_0$). The wave function of the particle is
$\psi(\phi)=N/(3+\sin(2\phi))$
a) What is the normalization constant?
b) What is the expectation value of $L_z$
c) Find the expectation value of the energy.

2 D polar coordinates:
$x=r_0*\cos⁡(φ)$,   $y=r_0*\sin⁡(φ)$
$dx=-r_0*\sin⁡(φ)*dφ,dy=+r_0*\cos⁡(φ)*dφ$
$∇^2 ψ=1/r*∂/∂r (r*∂ψ/∂r)+1/r^2 *(∂^2 ψ)/(∂φ^2 )=1/(r_0^2 )*(∂^2 ψ)/(∂φ^2 )$
a)
Normalization constant:
$1=N*∫_0^2π dφ/(3+\sin⁡(2φ))=N*π/√2$,that is $N=√2/π$
 b)   
$Lz=x*p_y-y*p_z=-ih_bar*(x*∂/∂y-y*∂/∂x)=-2ih_bar*∂/∂φ$
$L_z=<ψ|(L_z ) ̂ |ψ>$
$|(L_z |) ̂  ψ> =(-2i\hbar)(-√2/π*(2*\cos⁡(2φ))/(\sin⁡(2φ)+3)^2 )$
$L_z=(4i\hbar)/π^2 *∫_0^{2π} 1/((3+\sin⁡(2φ))*(2*\cos⁡(2φ))/((3+\sin⁡(2φ)^2 )*dφ=0$
c)
Particle is free to move. It means $V(r,φ)=0$. Hamiltonian is
$H=p^2/2m=-(\hbar^2)/2m*∇^2=-(\hbar^2)/2m*1/r_0^2 *∂^2/(∂φ^2 )$
$|H ̂ |ψ> =-(\hbar^2)/(2m r_0^2 )*√2/π*(2(6*\sin⁡(2φ)+\cos⁡(4φ)+3))/[\sin⁡(2φ)+3]^3$

$E=<ψ|H|ψ>= -(\hbar^2)/(2mr_0^2 )*4/π^2  ∫_0^2π (6 \sin⁡(2φ)+\cos⁡(4φ)+3)/(\sin⁡(2φ)+3)^4 *dφ=$
$=0.10413*(2\hbar^2)/(mr_0^2 π^2 )$

Reference
https://en.wikipedia.org/wiki/Particle_in_a_ring