Uniform convergence

Question

Does the given integral: $$\int_{0}^{+\infty}\frac{2x}{t^2}\space e^{-\left(t^2+\frac{x^2}{t^2} \right)} dt $$ Converge uniformly for x $\in \left]0, +\infty\right[$ ?

By bounding the integral I was able to show that the integral does converge uniformly for $x\in \left]a, b\right[$, where $a,b>0$. But is that true for the whole interval?

Answer 1

We assume $x>0$. By the change of variable, $$ u=\frac xt, \quad du= \frac {x}{t^2}\:dt, $$ one gets

$$ \begin{align} \int_{0}^{+\infty}\frac{2x}{t^2}\space e^{\large-\left(t^2+\frac{x^2}{t^2} \right)} dt& = 2\int_{0}^{+\infty}e^{\large -\left(u^2+\frac{x^2}{u^2} \right)} du \\\\&=2e^{-2x}\int_{0}^{+\infty}e^{\large -\left(u-\frac{x}{u}\right)^2} du \\\\&=\sqrt{\pi}\:e^{-2x} \end{align} $$

where we have used this related result.