Question

Integrate $\frac{\log x-1}{{\left(\log x\right)}^2}$ w.r.t. $x$.

Answer 1

Solution : Let $I=\int \frac{\log x-1}{{\left(\log x\right)}^2}dx$

$=\int \left(\frac{\log x}{{\left(\log x\right)}^2}-\frac{1}{{\left(\log x\right)}^2}\right)dx$

$=\int \left(\frac{1}{\log x}-\frac{1}{{\left(\log x\right)}^2}\right)dx$

$=\int \frac{1}{\log x}.1dx-\int \frac{1}{{\left(\log x\right)}^2}dx$

$=\frac{1}{\log x}.x-\int \frac{-1}{{\left(\log x\right)}^2}.\frac{1}{x}.xdx-\int \frac{1}{{\left(\log x\right)}^2}dx$

$=\frac{x}{\log x}+\int \frac{dx}{{\left(\log x\right)}^2}-\int \frac{dx}{{\left(\log x\right)}^2}$

$=\frac{x}{\log x}+c$