Mathematics - Indefinite Integration Question with Solution | TestHub

$$\begin{array}{rl}& \int \frac{(\log x-1{)}^2}{{\left(1+(\log x{)}^2\right)}^2}dx\\ & =\int \left[\frac{1}{\left(1+(\log x{)}^2\right)}-\frac{2\log x}{{\left(1+(\log x{)}^2\right)}^2}\right]dx\\ & =\int \left[\frac{e^t}{1+t^2}-\frac{2t{e}^t}{{\left(1+t^2\right)}^2}\right]dt\text{ put }\log x=t\Rightarrow dx=e^{t}dt\\ & \int e^t\left[\frac{1}{1+t^2}-\frac{2t}{{\left(1+t^2\right)}^2}\right]dt\\ & =\frac{e^t}{1+t^2}+c=\frac{x}{1+(\log x{)}^2}+c\end{array}$$