Hi
I want to integrate this integral and ask if my work is correct or not.
$\int^\infty_0 dx x^{\alpha-1} e^{-x} (a+bx)^{-\alpha}$
I want to integrate it by parts, so I have
$(a+bx)^{-\alpha} = v$
$-b\alpha(a+bx)^{-\alpha-1}dx = dv$
$x^{\alpha-1} e^{-x} dx = du$
$\Gamma(\alpha) = u$
now the integral becomes
$\left.\Gamma(\alpha)(a+bx)^{-\alpha}\right|_0^\infty + \int^\infty_0 \Gamma(\alpha) b\alpha(a+bx)^{-\alpha-1}dx = 0$
the problem is in integration by parts. Is it correct to put $\Gamma(\alpha) = u$. if it is not correct how can I compute this integral? please help.