Very Hard Integral (from int Coulomb) with -3/2 power + others

[pr0me7heu2]

Homework Statement

I am trying to directly calculate the electric field (using Coulomb) at some arbitrary point P(0,0,z). The charge is evenly distributed over the surface of a sphere (radius R, charge density σ). Here I use θ for the polar angle and p for the azimuthal angle.

I will leave out the messy details, but I know by symmetry only projection onto z-axis is relevant. I also determined the angle ψ (that between separation vector π and the z-axis) in terms of z,R,θ, and π.

Homework Equations

E_(alongz) = (4∏ε₀) ∫₀^2∏∫₀^∏ [σR^2 sinθ (z - Rcosθ)] / (R^2 + z^2 - 2Rzcosθ)^(3/2) dθ dp

The Attempt at a Solution

∫dp → 2∏
removing 2∏R^2σ constants out from integrand

∫₀^∏ [(z - Rcosθ)sinθ] / (R^2 + z^2 - 2Rzcosθ)^(3/2) dθ

using u-substitution:
u = cosθ du= -sinθ dθ
θ = 0 → u = 1
θ = ∏ → u =-1

and reversing the limits of integration gives (ignoring constants out front):

∫_-1¹ (z - Ru) / (R^2 + z^2 - 2Rzu)^(3/2) du​

(#1)

→according to solutions manual→ this works out to:

z^-2 [(z-R) / |z-R| - (-z-R) / |z+R|]​

(#2)

The manual says:

Integral can be done by partial fractions -- or look it up.

Does anyone have any idea how you would use partial fractions to go from (1) to (2)??

 

[haruspex]

I don't see how to use partial fractions here either, but how about a substitution t² = R²+z²-2Rzu?