Answer
$$-2\ln|\cos \sqrt x|+c.$$
Work Step by Step
Let $u=\sqrt x$; then $du=\frac{1}{2\sqrt x}dx$. Hence, we have
$$\int \frac{ \tan\sqrt x}{\sqrt x} d x=2\int \tan u \ du
\\
=2\int \frac{\sin u}{\cos u}du=-2\int \frac{d\cos u}{\cos u}\\
=-2\ln|\cos u|+c=-2\ln|\cos \sqrt x|+c.$$
