Integrate[(z-R Cos[th]) Sin[th]/(R^2 + z^2 - 2 R  z Cos[th])^(3/2),{th,0, Pi}]

Assuming[{z>0,R>0},Integrate[(z-R Cos[th]) Sin[th]/(R^2 + z^2 - 2 R  z Cos[th])^(3/2),{th,0, Pi}]]

Assuming[{z>0,R>0, z>R},Integrate[(z-R Cos[th]) Sin[th]/(R^2 + z^2 - 2 R  z Cos[th])^(3/2),{th,0, Pi}]]

Assuming[{z>0,R>0, z<R},Integrate[(z-R Cos[th]) Sin[th]/(R^2 + z^2 - 2 R  z Cos[th])^(3/2),{th,0, Pi}]]