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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.

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    No, it is not correct, since $\Gamma(\alpha)$ is a definite integral of that expression from $0$ to $\infty$, whereas for $u$ you need the indefinite integral of that expression.2011-02-27

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Just as an aside, this is the result from Wolfram Alpha

enter image description here

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    thank you. Now I can see it.2011-02-28