Justifying the existence of an improper integral

Question

function with 2 variables

can someone please help me with this

Answer 1

Upon making the change $t \to t\sqrt(x)$ the integral becomes: \begin{equation} F(x) = \frac{\pi}{4} \frac{\ln(x)}{\sqrt(x)}+\frac{1}{\sqrt(x)}\int_0^\infty\frac{\ln(t)}{1+t^2}dt \end{equation} The last integral is found to be $0$ by the change $t \to \frac{1}{t}$. So $F(x)$ exists and is defined for $x>0$ as: \begin{equation} F(x) = \frac{\pi}{4} \frac{\ln(x)}{\sqrt(x)} \end{equation}