calculate-0-e-x-2-1-x-2-dx-

Question Number 65676 by mathmax by abdo last updated on 01/Aug/19

c a l c u l a t e \int_{0}^{\infty} e^{- x^{2} - \frac{1}{x^{2}}} d x

Commented by ~ À ® @ 237 ~ last updated on 04/Aug/19

l e t n a m e d t h a t i n t e g r a l I . c h a n g e s x = \frac{1}{u} d x = \frac{- d u}{u^{2}}

I = \int_{0}^{\infty} e^{- \frac{1}{u^{2}} - u^{2}} \frac{d u}{u^{2}}

S o 2 I = \int_{0}^{\infty} (1 + \frac{1}{u^{2}}) e^{- u^{2} - \frac{1}{u^{2}}} d u

u^{2} + \frac{1}{u^{2}} = {(u - \frac{1}{u})}^{2} + 2

t h e n 2 I = e^{- 2} \int_{0}^{\infty} e^{- {(u - \frac{1}{u})}^{2}} (1 + \frac{1}{u^{2}}) d u

n o w l e t c h a n g e v = (u - \frac{1}{u}) d v = (1 + \frac{1}{u^{2}}) d u

s o 2 I = e^{- 2} \int_{- \infty}^{\infty} e^{- v^{2}} d v = e^{- 2} \sqrt{π}

f i n a l l y I = \frac{\sqrt{π}}{2 e^{2}}