I am looking for a better way to solve such questions, where we are asked to find $ \int_0 ^ \infty f(x) dx ,$
where $f(x)$ is a special function, such that we get the same integral when we replace $x$ by $\frac{1}{t}$ (with just variables changed).
Can we somehow exploit this fact, so as to develop some sort of short cut to solve such questions?
$\int f(x)dx =\int f(t)dt$ where t= $\frac{1}{x}$,
Examples of such class of questions are many, I have found two of them on MSE as of yet, this and this question.