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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!

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I edited your post. Is this the integral you meant? –  Mhenni Benghorbal Dec 19 '12 at 17:25
    
yes. thank you for your effort. –  user53719 Dec 19 '12 at 17:34
    
@user53719: You are welcome. –  Mhenni Benghorbal Dec 19 '12 at 17:38
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2 Answers

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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Thank you very much –  user53719 Dec 22 '12 at 13:50
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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).

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Thank you very much –  user53719 Dec 22 '12 at 13:50
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