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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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Since you are a new user, here are some tips. for better results try showing some work or explaining the steps you tried and where you got stuck. Also make sure to remember to accept an answer by clicking the check mark next to answers provided by other users, once you find one which satisfies your question. – MSEoris Dec 19 '12 at 17:13
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

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

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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