Question

Integrate $\frac{1}{x({\log }_{e}x{)}^n}$ with respect to $x$.

Answer 1

$I=\int \frac{1}{x({\log }_{e}x{)}^n}dx$
Put ${\log }_{e}x=t\Rightarrow \frac{1}{x}dx=dt$

$\Rightarrow I=\int \frac{1}{t^n}dx$
$=\int t^{-n}dx$
$=\frac{(t{)}^{-n+1}}{(-n+1)}+C$
$=\frac{({\log }_{e}x{)}^{1-n}}{1-n}+C$