Definite Integral to Infinity of Exponential of -a x^2
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Theorem
- $\ds \int_0^\infty e^{-a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi a}$
where $a$ is a strictly positive real number.
Proof
\(\ds \int_0^\infty e^{-a x^2} \rd x\) | \(=\) | \(\ds \int_0^\infty e^{-\paren {\sqrt a x}^2} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\sqrt a} \int_0^\infty e^{-t^2} \rd t\) | substituting $t = \sqrt a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \sqrt {\frac \pi a}\) | Integral to Infinity of $e^{-t^2}$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Exponential Functions: $15.72$