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Question Number 214360 by shunmisaki007 last updated on 06/Dec/24

$$\mathrm{Find}\underset{-\mathrm{\infty }}{\stackrel{\mathrm{\infty }}{\int }}{e}^{-{ax}^2}dx\mathrm{when}a\mathrm{is}\mathrm{constant}\mathrm{without}\mathrm{changing}\mathrm{the}\mathrm{coordinate}.$$

Answered by mathmax last updated on 06/Dec/24

$$=2{\int }_0^{\mathrm{\infty }}{e}^{-{ax}^2}dx\{a>0\}\sqrt{a}x=t$$$$=2{\int }_0^{\mathrm{\infty }}{e}^{-t^2}\frac{dt}{\sqrt{a}}=\frac{2}{\sqrt{a}}\times \frac{\sqrt{\pi }}{2}=\frac{\sqrt{\pi }}{\sqrt{a}}$$

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