I think my method is correct here, but i must of gone wrong somewhere (note i cannot use divergence theorem)

I have a vector field:

Where a is a constant and a sphere:

R is the radius.

Now i have to do a surface integral i.e. not gauss's divergence theorem of a volume integral. Though i did this in order to match my answers and i got

Though i haven't yet found a method to get near to an answer for the surface integral. I know:

Though in this situation dx, dy and dz are all present:

then i sub in n into the flux do the dot products, cancel the 1/|N| terms and get:

Now i introduce spherical coordinates whereby:

Now it looks a little messy but is still all doable - albeit a little long winded but im not allowed to use the divergence theorem.

To cut it short i get

Which is not the obvious and the one i got from divergence theorem:

Any help is much appreciated and thanks for reading