By choice of a suitable contour show that for $a>b>0$

$$\int_{0}^{2 \pi} \frac{\sin ^{2} \theta d \theta}{a+b \cos \theta}=\frac{2 \pi}{b^{2}}\left[a-\sqrt{a^{2}-b^{2}}\right]$$

Hence evaluate

$$\int_{0}^{1} \frac{\left(1-x^{2}\right)^{1 / 2} x^{2} d x}{1+x^{2}}$$

using the substitution $x=\cos (\theta / 2)$.