Find: $\displaystyle \int(\sqrt{\tan \mathrm{x}}+\sqrt{\cot \mathrm{x}}) d \mathrm{x}$.

\[\begin{aligned} & I=\int \frac{\sin \mathrm{x}+\cos \mathrm{x}}{\sqrt{\sin \mathrm{x} \cos \mathrm{x}}} d \mathrm{x} \\ & =\sqrt{2} \int \frac{\sin \mathrm{x}+\cos \mathrm{x}}{\sqrt{\sin 2 \mathrm{x}}} d \mathrm{x} \end{aligned} \] Put $\displaystyle \sin \mathrm{x}-\cos \mathrm{x}=\mathrm{t} \Rightarrow(\cos \mathrm{x}+\sin \mathrm{x}) \mathrm{dx}=\mathrm{dt}$ On squaring both sides, we get $\displaystyle 1-\sin 2 \mathrm{x}=\mathrm{t}^{2}$ \[I=\sqrt{2} \int \frac{1}{\sqrt{1-\mathrm{t}^{2}}} d \mathrm{t} \] $\displaystyle =\sqrt{2} \sin ^{-1}(\sin \mathrm{x}-\cos \mathrm{x})+\mathrm{C}$