TITLE: Solving the integral $\int\sqrt{\ln(x)}\,dx $
QUESTION [6 upvotes]: How to solve the following integral 
$$
\int\sqrt{\ln(x)}\,dx
$$
If the above integral is not solve-able, how to proof that the 
function $\sqrt{\ln(x)}$ is not integrable.

REPLY [3 votes]: Considering $$I=\int\sqrt{\log( x)}\ dx$$ change variable $$\sqrt{\log( x)}=t\implies x=e^{t^2}\implies dx=2te^{t^2}\,dt$$ which makes $$I=\int 2t^2e^{t^2}\,dt$$ Now, integration by parts $$u=t\implies u'=dt$$ $$v'=2te^{t^2}\,dt\implies v=e^{t^2}$$ $$I=te^{t^2}-\int e^{t^2}\,dt=te^{t^2} -\frac{\sqrt{\pi }}{2}  \text{erfi}(t)$$ where appears the imaginary error function.
Back to $x$, $$I=x \sqrt{\log (x)}-\frac{\sqrt{\pi }}{2}  \text{erfi}\left(\sqrt{\log (x)}\right)$$ which is real if $x\geq 1$.