Verifying that electrostatic potential satisfies Poisson's equation

[demonelite123]

i'm trying to understand how the electrosatic potential expressed as an integral satisfies poisson's equation. i know that i have to take the laplacian of both sides of (Eq 1.17) page 35 in Jackson.

i understood how jackson took the laplacian of \frac{1}{\sqrt{r^2 + a^2}} but after Eq 1.30 i am completely lost. he took a taylor expansion of p(x') around the point x' = x which i understand to second order is p(x) + (x' - x) * ∇p + (1/2)((x' - x) * ∇)²p.

but i have no idea how he calculated the integral over \theta and \phi to get the answer on the next line.

how would one go about this? is the process straightforward enough that jackson chooses to simply omit it in the text?

 

[Simon Bridge]

Don't have that text - I suspect it is similar to:
http://www.tat.physik.uni-tuebingen.de/~kokkotas/Teaching/Field_Theory_files/FT_course01.pdf

He shouldn't need to do the differentiation explicitly since the result should be independent of the coordinate system chosen.