Calculate the following real integrals by using contour integration. Justify your steps carefully.

(a)

$$I_{1}=\int_{0}^{\infty} \frac{x \sin x}{x^{2}+a^{2}} d x, \quad a>0$$

(b)

$$I_{2}=\int_{0}^{\infty} \frac{x^{1 / 2} \log x}{1+x^{2}} d x$$