Question

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

Answer 1

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

Put $\ln x=t,\frac{dx}{x}=dt$

$dx=xdt$

$dx=e^{t}dt$

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

We know that $\int e^x\left(f\right(x)+f(x\left)\right)dx=e^{x}f\left(x\right)+c$

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

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