Find the flux of F = <x^3/3, yz^2 + e^sqrt(xz), zy^2 + y + 2 + sin(x^3)> through upper hemisphere (z>=0) of x^2 + y^2 + z^2 = a^2

This is how I proceeded:

Let z = g(x,y)

=> g(x,y) = sqrt(a^2 - x^2 - y^2)

I found the partial differentials gx,gy

The flux

= integral F.<-gx,-gy,1> dA

After that I evaluated the dot product, replaced dA by r dr d(theta),

x by r cos(theta),

y by r sin(theta)

I took the limits of integrals as theta = 0 to 2pi, and r = 0 to a

But the integrals are becoming very complicated.

Is there a simpler way? Will finding flux through the disk which closes the hemisphere help?