I need some help in solving this integration:
$\int_0^{\gamma_t}K_0\left(2\sqrt{\frac{x}{a^2b^2c^2e}}\right)\text{d}x$
where, $K_0$ is modified Bessel function of second kind, $a,b,c,e$ are all constants.
Any help in this regard will be highly appreciated.
$$ \int_0^{\gamma t}{K_0}\left( 2\sqrt{\frac{x}{a^2b^2c^2e}} \right) \mathrm{d}x \\ x=\frac{a^2b^2c^2eu^2}{4},=\frac{a^2b^2c^2e}{2}\int_0^{\frac{2\sqrt{\gamma t}}{abc\sqrt{e}}}{uK_0}\left( u \right) \mathrm{d}u \\ =\frac{a^2b^2c^2e}{2}uK_1\left( u \right) |_{0}^{\frac{2\sqrt{\gamma t}}{abc\sqrt{e}}} \\ =abc\sqrt{e\gamma t}K_1\left( \frac{2\sqrt{\gamma t}}{abc\sqrt{e}} \right) $$