evaluate the integral $$I = \int_{-1}^1 \frac{(1-x^2)^{\frac{1}{2}}}{1+x^2} $$ using substitution $x=\cos(\theta)$ to show $$I=-\pi+\int_{-\pi}^{\pi}(1+cos^2(\theta))^{-1}d\theta$$ Then use $z=e^{i\theta}$ to write $$I = -\pi+\int g(z)dz$$ And hence evaluate I using the residue theorem.
I hope someone can help, i don't even know where to start. I subbed in $x=\cos(\theta)$, and i can't see how you change the limits from -1 & 1 to -pi & pi
Thanks in advance for any help :)