I found interesting for me integral's transform: $$\int_a^\infty {\frac{x}{\sqrt{x^2-a^2}}\int_x^\infty{K_1(z)}}dzdx = \int_a^\infty{\sqrt{z^2-a^2}K_1(z)}dz$$ where $K_1$ is the modified Bessel functions of the second kind.
Can someone explain this transform? And how can I get transform for similar integral, like this: $$\int_a^\infty {\int_x^\infty{K_1(z)}}dzdx$$
It comes from interchanging the integrals and noting that $\int_a^{z}\frac x {\sqrt {x^{2}-a^{2}}} dx=\sqrt {z^{2}-a^{2}}$. Note that $a<x<z$ so $x$ ranges from $a$ to $z$.
The second one is $\int_a^{\infty} (z-a) K_1(z)dz$.