I'm looking to evaluate the integral $$ I= \int_{-1}^{1} \frac {(1-x^2)^{1/2}}{(1+x^2)} dx $$ by considering the function $ f(z)= \frac {(z^2-1)^{1/2}}{z^2+1}$. Im told that using $ 0 < arg(z \pm 1) <2 \pi $ that we only need a branch cut between -1 and 1 of the real axis.
My method is to compute the integral $J $ for a circle of large radius (using the form of $ f(z)$ for large $z$) and then compute the integral $K$ around the cut, by a dumbbell contour. Then also accounting for the singularities at $i$ and $-i$ we can find the integral we want.
I was just wondering if this is a valid approach to the problem?