Question

Solve:

$\int \frac{\log x}{x^2}dx$

Answer 1

$I=\int \frac{logx}{x^2}dx$

Let $t=logx$

$\Rightarrow x=e^t$

$dt=\frac{1}{x}dx$

$I=\int t{e}^{-t}dt$

$I=t\frac{e^{-t}}{-1}-\int 1.\frac{e^{-t}}{-1}dt$

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

$I=-\frac{t}{e^t}-e^{-t}+c$

$I=\frac{-logx}{x}-\frac{1}{x}+c$

$I=-\frac{(logx+1)}{x}+c$.