
Integration
Could you help me with this?
By considering the integral of $\displaystyle e^{yx^2} $ over a suitable region of $\displaystyle \mathbb{R}^2 $, show that for $\displaystyle 0<a<b<\infty $,
$\displaystyle \int_{0}^{\infty} \frac{e^{ax^2}  e^{bx^2}}{x^2} dx = \sqrt{\pi} ( \sqrt{b}  \sqrt {a} ) $

the hint was already given, we have that $\displaystyle \frac{e^{ax^2}e^{bx^2}}{x^2}=\int_a^b e^{tx^2}\,dt,$ now reverse integration order (justified by Tonelli's Theorem since the integrand is non negative, becuase of $\displaystyle b>a$) and use the fact that $\displaystyle \int_{x\ge0}e^{x^2}\,dx=\frac{\sqrt\pi}2,$ hence, the result.