Answer
$$\int\ln\sqrt xdx=x\ln\sqrt x-\frac{x}{2}+C$$
Work Step by Step
$$A=\int\ln\sqrt xdx$$
We would choose $u=\ln\sqrt x$ and $dv=dx$
For $u=\ln\sqrt x$, according to Chain Rule, we have $$du=(\ln\sqrt x)'dx=\frac{d(\ln\sqrt x)}{d(\sqrt x)}\frac{d(\sqrt x)}{dx}dx=(\frac{1}{\sqrt x}\times\frac{1}{2\sqrt x})dx=\frac{1}{2x}dx$$
For $dv=dx$, then $v=x$
Apply Integration by Parts to A, we have $$A=uv−\int vdu$$ $$A=x\ln\sqrt x-\int x\frac{1}{2x}dx$$ $$A=x\ln\sqrt x-\int\frac{1}{2}dx$$ $$A=x\ln\sqrt x-\frac{x}{2}+C$$
