I need to integrate:

$$\int_p^{\infty} x\sqrt{x^2+a}K_0(bx)dx$$

but I can't do it. Following Gradshteyn and Ryzhik 7th Edition, formula 6.565.7 I can resolve it from 0 to infinity but not from p to infinity. I have tried to take off to 6.565.7 the same but between 0 and p but neither I know how to solve it. Any ideas for an exact result or an approximation?

  • $\begingroup$ Computing the integral from $p$ to $\infty$ for general $p$ is the same as finding an antiderivative for $x\sqrt{x^2+a}K_0(bx)$. So: can you do this special case: $\int \!x\sqrt {{x}^{2}+1}{{\rm K}_0\left(x\right)}{dx}$ $\endgroup$
    – GEdgar
    Oct 13, 2013 at 11:35
  • $\begingroup$ In fact, I was trying with a=1 doing a variable change but I can't. Sorry but I don't see the antiderivative of this. $\endgroup$ Oct 13, 2013 at 12:26


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