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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) – user 1618033 Aug 29 '12 at 8:56
thanks!! @Chris'sister – Santosh Linkha Aug 29 '12 at 8:56
up vote 9 down vote accepted

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 $\Bbb{R}$)

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