Prove-that-0-1-e-x-2-x-2-dx-

Question Number 194462 by Erico last updated on 08/Jul/23

Prove that {\int_{0}}^{+ \infty^{}} \frac{1 - e^{- x^{2}}}{x^{2}} d x = \sqrt{π}

Answered by mnjuly1970 last updated on 08/Jul/23

I = \int_{0}^{\infty} \frac{1 - e^{- x^{2}}}{x^{2}} d x \overset{?}{=} \sqrt{π}

I \overset{i . b . p}{=} {[- \frac{1}{x} (1 - e^{- x^{2}})]}_{0}^{\infty} + 2 \int_{0}^{\infty} e^{- x^{2}} d x

\underset{i n t e g r a l}{\overset{G a u s s}{=}} 2 (\frac{\sqrt{π}}{2}) = \sqrt{π}