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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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    @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, :-)2012-04-30

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