\[\begin{array}{l} I = \int\limits_0^{ + \infty } {\frac{{{e^{ - {x^2}}} - {e^{ - 2{x^2}}}}}{{{x^2}}}dx} \hfill \\ f\left( t \right) = \int\limits_0^{ + \infty } {\frac{{{e^{ - {x^2}}} - {e^{ - t{x^2}}}}}{{{x^2}}}dx} \Rightarrow f'\left( t \right) = \int\limits_0^{ + \infty } {{e^{ - t{x^2}}}dx} = \frac{{\sqrt \pi }}{{2\sqrt t }} \Rightarrow \hfill \\ f\left( t \right) = \sqrt {\pi \cdot t} + C \hfill \\ f\left( 1 \right) = \sqrt \pi + C = 0 \Leftrightarrow C = - \sqrt \pi \hfill \\ I = f\left( 2 \right) = \sqrt {2\pi } - \sqrt \pi . \hfill \\ \end{array} \]