Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 2 down vote accepted

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.

share|improve this answer
    
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? –  sidht Nov 30 '12 at 20:10
3  
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.