Solve the integral of function using parts. f(x)=(x^(-2))*(ln^2 x)

We'll recall the formula of integrating by
parts:

Int udv = u*v - Int
vdu

Let u = (ln x)^2 => du = 2*ln x
dx/x

Let dv = dx/x^2 => v =
-1/x

Int (ln x)^2 dx/x^2 = -(ln x)^2/x + 2 Int (ln x)
dx/x^2 (*)

We'll integrate by parts again the integral Int
(ln x) dx/x^2:

Let u = (ln x) => du =
dx/x

Let dv = dx/x^2 => v =
-1/x

Int (ln x) dx/x^2 = -(ln x)/x + Int
dx/x^2

Int (ln x) dx/x^2 = -(ln x)/x - 1/x + C
(**)

We'll replace (**) in
(*):

Int (ln x)^2 dx/x^2 = -(ln x)^2/x + 2*[-(ln x)/x -
1/x] + C

Int (ln x)^2 dx/x^2 = -[(ln x)^2 +
(ln (x^2)) + 2]/x + C