State the residue theorem.

By considering

$$\oint_{C} \frac{z^{1 / 2} \log z}{1+z^{2}} d z$$

with $C$ a suitably chosen contour in the upper half plane or otherwise, evaluate the real integrals

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

and

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

where $x^{1 / 2}$ is taken to be the positive square root.