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.
Integrals with special functions
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$\begingroup$
improper-integrals
hypergeometric-function
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0What are $\mu, s, P, \alpha$ ? – 2017-02-03
1 Answers
0
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).