Differentiation under integration

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.

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1 Answer

One of the problems with $$\int_0^\infty \frac{\sin(x)}{x}\ dx$$ is that it only converges conditionally, which makes differentiation under the integral quite delicate. Thus you might have tried $$\int_0^\infty \frac{\sin(ax)}{x}\ dx$$ where again taking the derivative of the integrand with respect to the parameter gets rid of the $x$ in the denominator, but this is not good because $\int_0^\infty \sin(ax)\ dx$ diverges. You want to put in a factor that improves the convergence, and $e^{-ax}$ does that.

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No, but I mean is there a way of examining the integrand and coming up with the placement of the extra factors inside. But from your response, I am guessing you mean to examine it's convergence for improper integrals? What about just regular definite integrals? –  Hawk Nov 30 '12 at 20:10
Basically what you want to do is try to come up with something that, when you take the derivative with respect to the parameter, will give you an easy integration. I don't know if there are really general guidelines. –  Robert Israel Nov 30 '12 at 21:03