Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|cite|improve this question
Nice question (+1) – OFFSHARING Aug 29 '12 at 8:56
thanks!! @Chris'sister – Santosh Linkha Aug 29 '12 at 8:56

1 Answer 1

up vote 8 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 $\Bbb{R}$)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.