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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.

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Tomarinator
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  • If I understand your question correctly, you are interested in $f$ such that

    $\int_0^\infty f(x)dx = -\int_\infty^0 f(1/t)\frac{1}{t^2}dt=\int_0^\infty \frac{f(1/t)}{t^2}$.

    and now you would like $f(t)=-f(1/t)/t^2$, to conclude that the integral must be zero (in accordance with your first link)?

     – Alex R. Apr 29 '12 at 17:23
  • $\int f(x)dx =\int f(t)dt$ where t= $\frac{1}{x}$, thats what I meant. – Tomarinator Apr 30 '12 at 14:58
  • Then you are not doing a proper substitution of variables. My apologies if that's not what were trying to do. – Tyler Hilton Apr 30 '12 at 15:03
  • @TylerHilton I have gotten the answer to those question by other means and "proper" substitution, but I noticed this one thing about these functions and I became curious as to whether we can come up with some sort of shortcut. I have a feeling that there is, and i am working on it, :-) – Tomarinator Apr 30 '12 at 15:36

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