I'm trying to solve this for a few days and no way. I should be able to do it for an exam this friday. It's part of an electric field calculation.

Integral over 1/|r-r'| over a sphere surface. r is the variable, r' is constant.

I'm attaching an image to show the problem.

Black is the problem itself, red is the solution (I need the way how to come to this), blue are my illustrations.

Anything without vector sign mean the length of the corresponding vector.

In my skript they suggest to chose the z-axis in direction of r' and to use polar coordinates.

Please help!

Also 1/|r-r'| can be rewritten with spherical functions Ylm, but I'm not sure if that helps in this problem (when I try this way, I'm stuck with the infinite summation over l, and they have different arguments so I can't really collapse them).