Could someone help me through this problem?
Let $C$ be an open (upper) semicircle of radius R withits center at the origin, and consider $$\displaystyle\int_{C} f(z)\, dz.$$ Let $\displaystyle f(z)=\frac{1}{z^{2}+a^{2}}$ for real $a>0$. Show that $$|f(z)|\leq \frac{1}{R^{2}-a^{2}}~\text{for }R>a,$$ and $$\left|\displaystyle\int_{C} f(z)\, dz \right|\leq \frac{\pi R}{R^{2}-a^{2}}~\text{for }R>a$$