Integrals with special functions

Question

Can someone help me to resolve this integral using some special functions, and showing me steps:

$$ \int_r^{\infty} \; (1- \frac{1}{(1+\mu sPx^{-\alpha})^{n}}) \, x dx $$

Many thanks in advance.

Answer 1

Hint:

By rescaling of the variable, you can bet rid of the constants. Then with $x=t^{-1/\alpha}$,

$$\int \left(1-\frac1{(1+x^{-\alpha})^n}\right)x\,dx=-\frac1\alpha \int t^{-1-2/\alpha}\left(1-\frac1{(1+t)^n}\right)dt$$

You have two terms: a power (which is easy) and a ratio

$$\int\frac{t^{a}}{(1+t)^b}dt.$$

The latter is solved as an incomplete Beta integral (https://en.wikipedia.org/wiki/Beta_function).