(First off I hope I am posting in the right location)

So I am pretty new to complex analysis, but I seem to have a handle on most contour integrals that can be solved by direct use of the residue theorem and I am getting stuck dealing with Branch cuts.

Particularly I am dealing with what I think is a frustrating integral.

Using contour integration I am suppose to deal with

$\displaystyle \int \!{\frac {1}{ \left( {a}^{2}+{x}^{2} \right) \sqrt {1-{x}^{2}}}}{

dx}

$

from -1 to 1

I figured out that I want to use a branch cut from -1 to 1 and a contour that goes counter-clockwise around the two branch points at -1 and 1 in the shape of a dumbbell (or dogbone) with the semi-circle ends near -1 and 1.

Any help would be great.

Thanks