Prove the wave function for dxy orbital

[chem1309]

Homework Statement

"The angular part of the wave function for the dxy orbital is (√(15/∏)/4)sin^2(θ)sin(2ϕ). Show that this expression corresponds to the dxy orbital"

Homework Equations

conversion of Cartesian to spherical coordinates:
r=√(x^2+y^2+z^2)
cosθ=z/r
tan(ϕ)=y/x

trig identity:
sin(2x)=2sinxcosx

normalization:
N^2∫ψ*ψdτ=1

dτ=r^2sinθdrdθdϕ

0≤r≤∞
0≤θ≤∏
0≤ϕ≤2∏

The Attempt at a Solution

in Cartesian coordinates dxy is represented as simply xy. I converted xy to spherical coordinates and manipulated the equation the relevant equations to get xy=(r/2)sin^2(θ)sin(2ϕ) as follows:

xy=rsincosϕrsinθsinϕ
xy=rsin^2(θ)cosϕsinϕ
xy=rsin^2(θ)sin(2ϕ)/2

Then I tried to normalize the equation, but I ended up with

∫r^3 from 0 to ∞, which goes to ∞/does not converge

and ∫sin2ϕ which equal zero.

 

[Nino MMP]

Do I need to use another identity to make the integral converge, or am I missing something entirely?