Question

Evaluate:$\int \frac{{(\ln x-1)}^2}{{(1+{\left(\ln x\right)}^2)}^2}dx$

Answer 1

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

Let $t=\ln x\Rightarrow dt=\frac{1}{x}dx$

$=\int {\left(\frac{t-1}{1+t^2}\right)}^2{e}^{t}dt$

$=\int \left(\frac{t^2+1-2t}{(1+t^2{)}^2}\right)e^{t}dt$

$=\int (\frac{1}{1+t^2}-\frac{2t}{(1+t^2{)}^2})e^{t}dt$

$=\int \frac{d}{dt}\left(\frac{e^t}{1+t^2}\right)dt$

$=\frac{e^t}{1+t^2}+c$

$=\frac{x}{1+(\ln x{)}^2}+c$