We have, $f(x) = – \frac{1}{{{x^2}}}f\left( {\frac{1}{x}} \right)$

$\therefore I = \int\limits_{\tan \alpha }^{\cot \alpha } { – \frac{1}{{{x^2}}}f\left( {\frac{1}{x}} \right)dx} $

Let, $\frac{1}{x} = t$

$ \Rightarrow – \frac{1}{{{x^2}}}dx = dt$

$\therefore I = \int\limits_{\cot \alpha }^{\tan \alpha } {f(t)dt} =-I $

$ \Rightarrow I = 0$