Find : \[\int \frac{\mathrm{x}^{-1}}{(\log \mathrm{x})^{2}-5 \log \mathrm{x}+4} d \mathrm{x} \]

Let $\displaystyle \log \mathrm{x}=\mathrm{t} ; \frac{1}{\mathrm{x}} \mathrm{dx}=\mathrm{dt}$ Given integral $\displaystyle =\int \frac{1}{\mathrm{t}^{2}-5 \mathrm{t}+4} d \mathrm{t}$=\int \frac{1}{\left(\mathrm{t}-\dfrac{5}{2}\right)^{2}-\left(\dfrac{3}{2}\right)^{2}} d \mathrm{t} =\frac{1}{3} \log \left|\frac{\mathrm{t}-4}{\mathrm{t}-1}\right|+C =\frac{1}{3} \log \left|\frac{\log \mathrm{x}-4}{\log \mathrm{x}-1}\right|+C $$