Hi i'm stuck on a problem and was wondering if anyone could help. The problem is as follows:

"It is required to evaluate

$\displaystyle L=\int_{- \ 1}^{1}\frac {({1-x^2})^\frac{1}{2}}{1+x^2}dx$,

using contour integration.

a) consider the function $\displaystyle f(z)=\frac {({z^2-1})^\frac{1}{2}}{z^2+1}$

(You are given that, with $\displaystyle -\pi<\arg(z\pm1)\leq\pi$, the only branch cut needed is the section [-1,1] of the real axis)

i)Find the value of $\displaystyle M=\oint f(z)dz$ around a circle of large radius.

ii)Relate M to the value of $\displaystyle N=\oint f(z)dz$ around the cut.

iii)Express N in terms of L, and hence evaluate L."

Thanks for reading.