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I need to find the following integration. $$ \int_0^a e^{-(N-1)x}(\sqrt{4x}K_{1}(\sqrt{4x}))^N $$ where $$ a>0, \quad N \geq 1 $$ Any help will be much appreciated. BR Frank

I need to compute the following integral. $$ \int_0^a e^{-bx}\sqrt{4(a-x)}K_1(\sqrt{4(a-x))}dx\,. $$ where $$ a>0$$ and $b$ can be greater than zero or less than zero but it is not a complex ...