If-0-e-x-2-dx-pi-2-then-prove-that-0-e-ax-2-dx-pi-4a-where-a-gt-0-

Question Number 43147 by rahul 19 last updated on 07/Sep/18

If \int_{0}^{\infty} e^{- x^{2}} d x = \frac{\sqrt{π}}{2},

then p r o v e t h a t \int_{0}^{\infty} e^{- a x^{2}} d x = \sqrt{\frac{π}{4 a}}

where a > 0 .

Commented by MrW3 last updated on 07/Sep/18

l e t t = \sqrt{a} x

d t = \sqrt{a} d x

d x = \frac{d t}{\sqrt{a}}

\int_{0}^{\infty} e^{- {a x}^{2}} d x

= \int_{0}^{\infty} e^{- t^{2}} \frac{d t}{\sqrt{a}}

= \frac{1}{\sqrt{a}} \int_{0}^{\infty} e^{- t^{2}} d t

= \frac{1}{\sqrt{a}} \times \frac{\sqrt{π}}{2}

= \sqrt{\frac{π}{4 a}}

Commented by rahul 19 last updated on 07/Sep/18

thank you sir ��