Definite integration question

This isn't a homework problem, but something I've been puzzled by for a
while.

I'm trying to find an analytic form of the surface integral:

\int_S{ (l + h cos (phi))^-1 dh dphi }

over the circle h=0 to h0, phi=0 to 2*pi. The one bit of information I
don't know how to represent is that l>h. There's a pole in the system at
l=+/-h, so perhaps I need to include this somehow.

Doing this numerically works, and it converges.

My textbook suggests the substitution g=tan(0.5*x), but evaluating this
returns the answer:

[insert subs result]

which evaluates to zero. It's not as if we're introducing a pole at
g=tan(pi) since the integral will still evaluate.

[look at wikipedia closed form]
.

Relevant Pages

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