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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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

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