Answer :
[tex]I=\sqrt{tan\ x}+\sqrt{cot\ x},\ \ \ Let\ tan\ x=t^2\\\\2t\ dt=sec^2x\ dx=(1+t^2)dx\\\\ I= \int\limits^{}_{} {(t+\frac{1}{t})\frac{2t}{1+t^2}} \, dt \\\\= \int\limits^{}_{} {\frac{(t^2+1)}{t}\frac{2t}{1+t^2}} \, dt\\\\= \int\limits^{}_{} {2} \, dt \\\\=2t+K\\\\=2\sqrt{tan\ x}+K\\[/tex]