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.