\int _{0}^{2}{\frac{dx}{x\sqrt\ln x}} = \lim_{\epsilon \to 0+}{\int _{\epsilon}^{2}{\frac{dx}{x\sqrt\ln x}}} = \lim_{\epsilon \to 0+} 2\sqrt{ln x}{\line(0 ,90 ){20}_\epsilon^2} = \lim_{\epsilon \to 0+} 2\sqrt{ln 2} + 2\sqrt{ln  \epsilon} = \lim_{\epsilon \to 0+} 2\sqrt{ln 2} + 2\sqrt{ln 0}