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How do I evaluate the following definite integral$ \int_0^\infty e^{- \left ( x - \frac a x \right )^2} dx $

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    Nice question (+1)2012-08-29

2 Answers 2

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I will assume that $a > 0$. Let $y = \frac{a}{x}$. Then

$I := \int_{0}^{\infty}e^{-\left(x-\frac{a}{x}\right)^2}\;dx = \int_{0}^{\infty}\frac{a}{y^2} \, e^{-\left(y-\frac{a}{y}\right)^2}\;dy.$

Thus we have

$2I = \int_{0}^{\infty}\left(1 + \frac{a}{x^2}\right) e^{-\left(x-\frac{a}{x}\right)^2}\;dx. $

Now by the substitution $t = x - \frac{a}{x}$,

$2I = \int_{-\infty}^{\infty} e^{-t^2} \; dt = \sqrt{\pi}.$

Therefore $I = \frac{\sqrt{\pi}}{2}$.

(You can see that this generalizes to any integrable even function on $\Bbb{R}$)