Evaluate : \[\int_{1}^{\mathrm{e}} \frac{1}{\sqrt{4 \mathrm{x}^{2}-(\mathrm{x} \log \mathrm{x})^{2}}} d \mathrm{x} \]

$$\begin{aligned} & \text { Let } \mathrm{I}=\int_{1}^{\mathrm{e}} \frac{1}{\sqrt{4 \mathrm{x}^{2}-(\mathrm{x} \log \mathrm{x})^{2}}} \mathrm{dx} \\ & =\int_{1}^{\mathrm{e}} \frac{1}{\mathrm{x} \sqrt{4-(\log \mathrm{x})^{2}}} \mathrm{dx} \quad\left[\text { Let } \log \mathrm{x}=\mathrm{t} \Rightarrow \frac{1}{\mathrm{x}} \mathrm{dx}=\mathrm{dt}\right] \\ & =\int_{0}^{1} \frac{d \mathrm{t}}{\sqrt{4-\mathrm{t}^{2}}} \\ & =\left.\sin ^{-1} \frac{\mathrm{t}}{2}\right|_{0} ^{1}=\frac{\pi}{6} \end{aligned} $$