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I have been trying to evaluate the following integral:

$$I = \int_{a}^{\infty}dx \frac{x^s}{(1+x)^{n}}$$

If $a=0$, then this is the Mellin transform of $\frac{1}{(1+x)^n}$. However, suppose $a \neq 0$. Is there a strategy to approach this problem? Wolfram alpha expresses this integral in terms of the incomplete beta function. Is there an analytical way to understand this result?

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  • $\begingroup$ This is typical, actually, of nontrivial integrals: the best that can happen is that they are expressible in terms of a standard transcendental function, for instance a hypergeometric function. If you are looking for something simpler then you might be out of luck. Getting a rational expression involving Gamma functions is actually pretty nice, in comparison. $\endgroup$
    – David Roberts ♦
    Commented Oct 3, 2021 at 6:07

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As a direct answer to your last question, substituting $x=(1-t)/t$ in your integral gives $$\int_0^{1/(1+a)}(1-t)^st^{n-s-2}\,dt$$ which is the standard definition of an incomplete beta integral. That is a perfectly good answer, you can't expect an expression in terms of elementary functions if all your parameters are general.

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