Tip for calculating an integral

Question

I need to calculate this integral and have no idea what technique to use. Can anyone give me a hint?

$$ \int_{a}^{+\infty} \dfrac{1}{r}\bigg(3\dfrac{\cos(kr)}{(kr)^2}-3\dfrac{\sin(kr)}{(kr)^3}+\dfrac{\sin(kr)}{kr}\bigg) dr, \ \mbox{where} \ a,k > 0 \ \mbox{are constant} $$

I appreciate the help

Answer 1

Here is how to compute the antiderivative "manually", step by step:

Then we just need to calculate the improper integral as x approaches$+\infty$.

Answer 2

Use:

$$\int\limits_{r=a}^\infty \frac{\sin (r)}{r^n}\ dr = \frac{a^{2-n} \, _1F_2\left(1-\frac{n}{2};\frac{3}{2},2-\frac{n}{2};-\frac{a^2}{4}\right)}{n-2}+\cos \left(\frac{\pi n}{2}\right) \Gamma (1-n)$$

where $F$ is a hypergeometric function.