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I would like to find the answer for the following integral

$\int x\ln(x)K_0(x) dx $

where $K_0(x)$ is the modified Bessel function of the second kind and $\ln(x)$ is the natural-log. Do you have any ideas how to find?

Thanks in advance!

  • 0
    @user53719: You are welcome.2012-12-19

2 Answers 2

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Use integration by parts and the fact that $\int x K_0(x)dx = -x \frac{d}{dx}K_0(x)=-xK'_0(x)$

$ \int x\ln(x)K_0(x)\,dx = -x\ln(x)K'_0(x) - \int (-x K'_0(x))(\frac{1}{x}) \, dx =\dots. $

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Here's what Mathematica found:

Mathematica graphics

Looks like an integration by parts to me (combined with an identity for modified Bessel functions).