Evaluate the real integral

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

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

What is the value of

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