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.