Is there a way to simplify integrals of the form $\int_{-\infty}^\infty \frac{x \sin(kx)}{(x^2+b^2)^{3/2}}dx$?

Question

Is there a way to simplify integrals in the form: $$\int_{-\infty}^\infty \frac{x \sin(kx)}{(x^2+b^2)^{3/2}}dx$$ and $$\int_{-\infty}^\infty \frac{x \sin(kx)}{(x^2+b^2)^{5/2}}dx$$ or am I stuck with numerical methods?

Thank you.

Answer 1

Assuming $k>0$ and $b>0$, considering $$I=\int_{-\infty}^\infty \frac{x\sin(kx)}{(x^2+b^2)^{3/2}}\,dx$$first, let $x=bt$ to make $$I=\frac 1 b\int_{-\infty}^\infty \frac{t \sin (b k t)}{\left(t^2+1\right)^{3/2}}\,dt=2k\,K_0(bk)$$ Do the same for the other one and get $$J=\int_{-\infty}^\infty \frac{x\sin(kx)}{(x^2+b^2)^{5/2}}\,dx=\frac {2k^2}3 K_1(b k)$$