I know there are other similar questions on this topic, but I am wondering if there is some sort of general method to pick where to place my parameters in the integral?
For instance, $\int_{0}^{\infty} \frac{\sin(x)}{x} dx\;,$ how does one know to add a parameter like $e^{-ax}$ to $\int_{0}^{\infty}e^{-ax} \frac{\sin(x)}{x} dx\;?$
Wikiapedia has a non-exhaustive list of problems like this: see link.
But I do not know how on earth they come up with those crazy methods.