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I am getting stuck in integrating the below equation that includes modified Bessel function of second kind:

$$\int_0^{\frac{y}{\eta}}K_0\left(\frac{|x|}{\sqrt{\sigma_a^2\sigma_b^2}}\right)\mathrm{d}x$$

Any help in this regard is highly appreciated.

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Make the appropriate change of variable and $$\int K_0(t)\,dt=\frac{\pi}{2} \, t \,(\pmb{L}_{-1}(t) K_0(t)+\pmb{L}_0(t) K_1(t))$$ where appear the modified Struve functions.

Learn about them since they appear in most integrals of Bessel functions.

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