Let ∫ 1 - ln x x^ 2 dx = f(x), for all positive x . If f (e) =...

$f\left(x\right) = ∫ \dfrac{1}{x^{2}}dx - ∫ ln x ⋅ \dfrac{1}{x^{2}}dx$
$= - \dfrac{1}{x} - \left\{ln x\left(- \dfrac{1}{x}\right) + ∫ \dfrac{1}{x^{2}}dx\right\}$
$= - \dfrac{1}{x} + \dfrac{ln x}{x} + \dfrac{1}{x} + c = \dfrac{ln x}{x} + c$
Now, $f\left(e\right) = \dfrac{1}{e} ⇒ c = 0$

Hence, $f\left(2\right) + f\left(4\right) = \dfrac{\ln 2}{2} + \dfrac{\ln 4}{4} = \ln 2$