Integral xsinx limits 0 to infinity

[skateboarding]

Homework Statement

So I'm trying to evaluate the following integral:

$$4\pi r^2{\int_0}^\infty r^2\frac{\sin{sr}}{sr}dr$$

which after canceling out one of the r's, gives an integral similar to that of xsinx.
I need to show that this integral vanishes for all values of s that are not 0. In other words, this integral has a non-zero value only when s is equal to zero. This leads me to think that this integral is some sort of a delta function.

By integration by parts I get the following:

$$\frac{4\pi}{s^3}(\sin{sr} -sr\cos{sr}){\mid_0}^\infty$$

I've tried to play around with trig identities when evaluating the limits of integration, but can't seem to get anything that doesn't involve trig functions to get it to converge, since the limit of cos(x) or sin(x) as x goes to infinity is undefined due to it's oscillatory nature. I have also tried evaluating this integral by first using euler's formula to convert the sin function into powers of e, but this results in similar problems when evaluating the limits. Any suggestions would be appreciated. This problem comes from x-ray scattering, so maybe someone familiar with this subject has seen this.

Homework Equations

The Attempt at a Solution

 

[stringy]

Your antiderivative looks correct. It appears that the integral doesn't converge though. No amount of rewriting (via trig identities, algebra, etc.) is going to force an integral that doesn't converge to converge.

EDIT: It's been awhile since I've taken any physics courses, but I remember in E&M always coming across integrals that blew up. That was usually a sign I screwed up earlier in my calculations.